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As-salamu alaykum
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As-salāmu ʿalaykum (Arabic: السَّلَامُ عَلَيْكُمْ [asːaˈlaːmu ʕaˈlaikum]) is a greeting in Arabic that means "peace be upon you". The greeting is a standard salutation among Muslims, whether socially or within worship and other contexts. [1] The typical response to the greeting is waʿalaykumu as-salām (وَعَلَيْكُم السَّلَام [waʕaˈlaikumu sːaˈlaːm]; "and upon you, peace").
This greeting appears in greatly abbreviated forms in many languages from Malagasy to Urdu as some variant of salām (سَلَام; cf. Persian سلام [sæˈlɒːm]).
Contents
[hide]
1 Grammatical variants
2 In Islam
3 Usage by non-Arabic speakers
4 See also
5 References
6 External links
Grammatical variants[edit]
The expression uses the second person plural masculine, even when used to address one person. It may be modified by choosing the appropriate enclitic pronoun to address a person in the masculine and feminine singular form, the dual form, or the feminine plural form. The conjugations are as follow (note: according to the standard pronunciation rules of Classical Arabic, the last short vowel in each word is not pronounced unless it is followed by another word):
masculine singular as-salāmu ʿalayka (عَلَيْكَ)
feminine singular: as-salāmu ʿalayki (عَلَيْكِ)
dual: As-salāmu ʿalaykumā (عَلَيْكُمَا)
feminine plural: As-salāmu ʿalaykunna (عَلَيْكُنَّ)
A third-person variant, ʿalayhi as-salām "peace be upon him", is used in reference to prophets.
In Islam[edit]
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It is also preferred to use the greeting when arriving and also while leaving. It was reported that Abu Hurairah said "When one of you joins a gathering, let him say 'Peace'. When he wants to get up and leave, let him say 'Peace'. The former is not more important than the latter" (Hasan hadith reported in Jāmiʿ at-Tirmidhi).[2]
According to hadith, Muhammad was asked who should begin the greeting and he said, "The one who is riding should greet the one who is walking and the one who is walking should greet the one who is sitting and the smaller group should greet the larger group" (Ṣaḥīḥ al-Bukhārī, 6234; Muslim, 2160).[3]
It is also stated that one should give the Salam greeting upon entering a house. This is based upon a verse of the Quran: "But when you enter the houses, greet one another with a greeting from God blessed and good" (An-Nur 24:61).[4]
Shortening the greeting to acronyms, such as A.S., As'kum (in Malaysia), or AsA is becoming common amongst Internet users in chat rooms and by people using SMS. This trend is similar to writing (S) or SAWS in place of ṣallá l-Lāhi ʿalayhi wa s-salām.
Usage by non-Arabic speakers[edit]
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (May 2013) (Learn how and when to remove this template message)
In Afghanistan, Azerbaijan, and Iran, Salām is used.
In Albania and Kosovo, a dimunitive form in the Albanian language, Selamun Alejkem or Selamun Alejqum is rarely used, the 'q' being a voiceless palatal stop typical of Balkan Turkish and Thracian Turkish phonology.[5]
In Amharic, the native Amharic term Selam is used in place of Tadias, which is the equivalent of "What's up".
In Turkey and Kazakhstan, many religious people use this statement and shake hands and it is the same for saying "goodbye"; more secular and non-religious people say Selam as an equivalent to "Hello" or "Hi". However, many Turks pronounce it differently as "Selamün aleyküm".
In Pakistan, the greeting is also associated with shaking right hands and is also often accompanied with a hug when meeting infrequently (only between the same gender). In some places, people put a hand on their heart as they shake your hand and greet. Also, the full greeting is preferred versus the shorter greeting, "salam".
In India, the greeting mostly among Muslims is accompanied by raising the right hand to the chest (arz hai "regards"; adaab "respect") or a simple handshake or hug, the shorter greeting "Salam" is used in informal situations. Goodbye is supplanted by a "Khuda hafiz" (secular/less formal or to an acquaintance) or "Allah hafiz" (less secular/generally to strangers, formal), both of which mean "May God keep you safe".
In Bangladesh, Assalamu alaykum is a simple greeting.
In Uzbekistan, Assalomu aleykum is used as an informal greeting.
In Indonesia, the greeting is usually[citation needed] accompanied with a kind of two-handed "handshake", whereby the shaker's palms remain closed, and the fingers alone open to admit the other's proffered hand – which briefly touches the proffered's fingers or fingertips alone. In this way more adherent males and females may greet though touching – but remain true to the Islamic or cultural teachings forbidding physical contact between the genders. Occasionally, the right hand will touch the left breast or heart area after this.[citation needed] In Indonesia's Javanese/Sasak culture, a remnant of feudalism is retained, where an elder's proffered right hand is taken and pressed briefly against the forehead. Some may instead briefly kiss the hand or the main ring. This is very common for young children to greet older relatives (of their parents' age, though, on occasion, if very polite children, younger).
In Chechnya and other parts of Caucasus, Salam Aleykum is used to say hi.
In Senegal which has a majority of Sufi Islam it is a common greeting. Spelled and pronounced in Wolof: "(a)sala maaleykum", and the reply "maa lekum salaam."
In Xinjiang, China, "Essalam eleykum" is used as a greeting by Uyghurs, and the reply is "We'eleykum essalam".
See also[edit]
Islam portal
Names of God in Islam
Pax (liturgy)
Salam Pax
Shalom aleichem
Š-L-M
References[edit]
Jump up ^ ""As-Salaamu-Alaikum" and "Wa-Alaikum-as-Salaam"". Ccnmtl.columbia.edu. Retrieved 2013-07-27.
Jump up ^ Sheikh Muhammad Salih Al-Munajjid. "Is it mustahabb for one who gets up to leave a gathering to say salaam to those who are still sitting?". Islamqa.info.
Jump up ^ "As Salaamu Alaikom?". Archived from the original on 2010-11-20.
Jump up ^ "Surat An-Nur [24:61] - The Noble Qur'an - القرآن الكريم". Quran.com. Retrieved 2013-07-27.
Jump up ^ https://humstatic.uchicago.edu/slavic/archived/papers/Friedman-TurkEncyc.pdf
External links[edit]
Audio clip for Salam
A brief illustrated guide to understanding Islam
LivingHalal.com Islamic audio project
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Islamic Greetings
General
As-salamu alaykum
Wa alaykumu s-salam
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Law of large numbers
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An illustration of the law of large numbers using a particular run of rolls of a single die. As the number of rolls in this run increases, the average of the values of all the results approaches 3.5. While different runs would show a different shape over a small number of throws (at the left), over a large number of rolls (to the right) they would be extremely similar.
Part of a series on Statistics
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In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
The LLN is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the law only applies (as the name indicates) when a large number of observations is considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy).
Contents
1 Examples
2 History
3 Forms
3.1 Weak law
3.2 Strong law
3.3 Differences between the weak law and the strong law
3.4 Uniform law of large numbers
3.5 Borel's law of large numbers
4 Proof of the weak law
4.1 Proof using Chebyshev's inequality assuming finite variance
4.2 Proof using convergence of characteristic functions
5 See also
6 Notes
7 References
8 External links
Examples[edit]
For example, a single roll of a fair, six-sided dice produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability. Therefore, the expected value of a single die roll is
1 + 2 + 3 + 4 + 5 + 6 6 = 3.5 {\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5}
According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the precision increasing as more dice are rolled.
It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. For a Bernoulli random variable, the expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.)) is precisely the relative frequency.
For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1/2. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 1/2. In particular, the proportion of heads after n flips will almost surely converge to 1/2 as n approaches infinity.
Although the proportion of heads (and tails) approaches 1/2, almost surely the absolute difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number, approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows.
History[edit]
Diffusion is an example of the law of large numbers, applied to chemistry. Initially, there are solute molecules on the left side of a barrier (magenta line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container.
Top: With a single molecule, the motion appears to be quite random.
Middle: With more molecules, there is clearly a trend where the solute fills the container more and more uniformly, but there are also random fluctuations.
Bottom: With an enormous number of solute molecules (too many to see), the randomness is essentially gone: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. In realistic situations, chemists can describe diffusion as a deterministic macroscopic phenomenon (see Fick's laws), despite its underlying random nature.
The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials.[1] This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli.[2] It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713. He named this his "Golden Theorem" but it became generally known as "Bernoulli's Theorem". This should not be confused with Bernoulli's principle, named after Jacob Bernoulli's nephew Daniel Bernoulli. In 1837, S.D. Poisson further described it under the name "la loi des grands nombres" ("The law of large numbers").[3][4] Thereafter, it was known under both names, but the "Law of large numbers" is most frequently used.
After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev,[5] Markov, Borel, Cantelli and Kolmogorov and Khinchin. Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true.[6][7] These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law, in reference to two different modes of convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.[6]
Forms[edit]
Two different versions of the law of large numbers are described below; they are called the strong law of large numbers, and the weak law of large numbers. Stated for the case where X1, X2, ... is an infinite sequence of i.i.d. Lebesgue integrable random variables with expected value E(X1) = E(X2) = ...= µ, both versions of the law state that – with virtual certainty – the sample average
X ¯ n = 1 n ( X 1 + ⋯ + X n ) {\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})}
converges to the expected value
X ¯ n → μ for n → ∞ , {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\,\to \,\mu \qquad {\textrm {for}}\qquad n\to \infty ,\\{}\end{matrix}}}
(law. 1)
(Lebesgue integrability of Xj means that the expected value E(Xj) exists according to Lebesgue integration and is finite. It does not mean that the associated probability measure is absolutely continuous with respect to Lebesgue measure.)
An assumption of finite variance Var(X1) = Var(X2) = ... = σ2 < ∞ is not necessary. Large or infinite variance will make the convergence slower, but the LLN holds anyway. This assumption is often used because it makes the proofs easier and shorter.
Mutual independence of the random variables can be replaced by pairwise independence in both versions of the law.[8]
The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see Convergence of random variables.
Weak law[edit]
Simulation illustrating the law of large numbers. Each frame, a coin that is red on one side and blue on the other is flipped, and a dot is added in the corresponding column. A pie chart shows the proportion of red and blue so far. Notice that while the proportion varies significantly at first, it approaches 50% as the number of trials increases.
The weak law of large numbers (also called Khinchin's law) states that the sample average converges in probability towards the expected value[9]
X ¯ n → P μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 2)
That is, for any positive number ε,
lim n → ∞ Pr ( | X ¯ n − μ | > ε ) = 0. {\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |>\varepsilon \,\right)=0.}
Interpreting this result, the weak law states that for any nonzero margin specified, no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin.
As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in the series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by Chebyshev as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of the average of the first n values goes to zero as n goes to infinity.[7] As an example, assume that each random variable in the series follows a Gaussian distribution with mean zero, but with variance equal to 2 n / log  ( n + 1 ) . {\displaystyle 2n/\log(n+1).} At each stage, the average will be normally distributed (as the average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is asymptotic to n 2 / log  n {\displaystyle n^{2}/\log n} . The variance of the average is therefore asymptotic to 1 / log  n {\displaystyle 1/\log n} and goes to zero.
An example where the law of large numbers does not apply is the Cauchy distribution. Let the random numbers equal the tangent of an angle uniformly distributed between −90° and +90°. The median is zero, but the expected value does not exist, and indeed the average of n such variables has the same distribution as one such variable. It does not tend toward zero as n goes to infinity.
There are also examples of the weak law applying even though the expected value does not exist. See #Differences between the weak law and the strong law.
Strong law[edit]
The strong law of large numbers states that the sample average converges almost surely to the expected value[10]
X ¯ n → a.s. μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\bar {X}}_{n}\ {\xrightarrow {\text{a.s.}}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 3)
That is,
Pr ( lim n → ∞ X ¯ n = μ ) = 1. {\displaystyle \Pr \!\left(\lim _{n\to \infty }{\bar {X}}_{n}=\mu \right)=1.}
What this means is that as the number of trials n goes to infinity, the probability that the average of the observations is equal to the expected value will be equal to one.
The proof is more complex than that of the weak law.[11] This law justifies the intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average".
Almost sure convergence is also called strong convergence of random variables. This version is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). The strong law implies the weak law but not vice versa, when the strong law conditions hold the variable converges both strongly (almost surely) and weakly (in probability). However the weak law may hold in conditions where the strong law does not hold and then the convergence is only weak (in probability).[clarification needed]

The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem.
The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for the average to converge almost surely on something (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that).[12]
If the summands are independent but not identically distributed, then
X ¯ n − E  [ X ¯ n ] → a.s. 0 , {\displaystyle {\bar {X}}_{n}-\operatorname {E} {\big [}{\bar {X}}_{n}{\big ]}\ {\xrightarrow {\text{a.s.}}}\ 0,}
provided that each Xk has a finite second moment and
∑ k = 1 ∞ 1 k 2 Var  [ X k ] < ∞ . {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} [X_{k}]<\infty .}
This statement is known as Kolmogorov's strong law, see e.g. Sen & Singer (1993, Theorem 2.3.10).
An example of a series where the weak law applies but not the strong law is when Xk is plus or minus k / log  log  log  k {\displaystyle {\sqrt {k/\log \log \log k}}} (starting at sufficiently large k so that the denominator is positive) with probability 1/2 for each.[12] The variance of Xk is then k / log  log  log  k . {\displaystyle k/\log \log \log k.} Kolmogorov's strong law does not apply because the partial sum in his criterion up to k=n is asymptotic to log  n / log  log  log  n {\displaystyle \log n/\log \log \log n} and this is unbounded.
If we replace the random variables with Gaussian variables having the same variances, namely k / log  log  log  k , {\displaystyle {\sqrt {k/\log \log \log k}},} then the average at any point will also be normally distributed. The width of the distribution of the average will tend toward zero (standard deviation asymptotic to 1 / 2 log  log  log  n {\displaystyle 1/{\sqrt {2\log \log \log n}}} ), but for a given ε, there is probability which does not go to zero with n that the average sometime after the nth trial will come back up to ε. Since this probability does not go to zero[clarification needed], it must have a positive lower bound p(ε), which means there is a probability of at least p(ε) that the average will attain ε after n trials. It will happen with probability p(ε)/2 before some m which depends on n. But even after m, there is still a probability of at least p(ε) that it will happen. (This seems to indicate that p(ε)=1 and the average will attain ε an infinite number of times.)
Differences between the weak law and the strong law[edit]
The weak law states that for a specified large n, the average X ¯ n {\displaystyle {\overline {X}}_{n}} is likely to be near μ. Thus, it leaves open the possibility that | X ¯ n − μ | > ε {\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon } happens an infinite number of times, although at infrequent intervals. (Not necessarily | X ¯ n − μ | ≠ 0 {\displaystyle |{\overline {X}}_{n}-\mu |\neq 0} for all n).
The strong law shows that this almost surely will not occur. In particular, it implies that with probability 1, we have that for any ε > 0 the inequality | X ¯ n − μ | < ε {\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon } holds for all large enough n.[13]
The strong law does not hold in the following cases, but the weak law does.[14][15][16]
1. Let X be an exponentially distributed random variable with parameter 1. The random variable sin  ( X ) e X X − 1 {\displaystyle \sin(X)e^{X}X^{-1}} has no expected value according to Lebesgue integration, but using conditional convergence and interpreting the integral as a Dirichlet integral, which is an improper Riemann integral, we can say:
E ( sin  ( X ) e X X ) = ∫ 0 ∞ sin  ( x ) e x x e − x d x = π 2 {\displaystyle E\left({\frac {\sin(X)e^{X}}{X}}\right)=\ \int _{0}^{\infty }{\frac {\sin(x)e^{x}}{x}}e^{-x}dx={\frac {\pi }{2}}}
2. Let x be geometric distribution with probability 0.5. The random variable 2 X ( − 1 ) X X − 1 {\displaystyle 2^{X}(-1)^{X}X^{-1}} does not have an expected value in the conventional sense because the infinite series is not absolutely convergent, but using conditional convergence, we can say:
E ( 2 X ( − 1 ) X X ) = ∑ 1 ∞ 2 x ( − 1 ) x x 2 − x = − ln  ( 2 ) {\displaystyle E\left({\frac {2^{X}(-1)^{X}}{X}}\right)=\ \sum _{1}^{\infty }{\frac {2^{x}(-1)^{x}}{x}}2^{-x}=-\ln(2)}
3. If the cumulative distribution function of a random variable is
1 − F ( x ) = e 2 x ln  ( x ) , x ≥ e {\displaystyle 1-F(x)={\frac {e}{2x\ln(x)}},x\geq e}
F ( x ) = e − 2 x ln  ( − x ) , x ≤ − e {\displaystyle F(x)={\frac {e}{-2x\ln(-x)}},x\leq -e}
then it has no expected value, but the weak law is true.[17][18]
Uniform law of large numbers[edit]
Suppose f(x,θ) is some function defined for θ ∈ Θ, and continuous in θ. Then for any fixed θ, the sequence {f(X1,θ), f(X2,θ), …} will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E[f(X,θ)]. This is the pointwise (in θ) convergence.
The uniform law of large numbers states the conditions under which the convergence happens uniformly in θ. If[19][20]
Θ is compact,
f(x,θ) is continuous at each θ ∈ Θ for almost all xs, and measurable function of x at each θ.
there exists a dominating function d(x) such that E[d(X)] < ∞, and
‖ f ( x , θ ) ‖ ≤ d ( x ) for all θ ∈ Θ . {\displaystyle \left\|f(x,\theta )\right\|\leq d(x)\quad {\text{for all}}\ \theta \in \Theta .}
Then E[f(X,θ)] is continuous in θ, and
sup θ ∈ Θ ‖ 1 n ∑ i = 1 n f ( X i , θ ) − E  [ f ( X , θ ) ] ‖ → a . s . 0. {\displaystyle \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta )-\operatorname {E} [f(X,\theta )]\right\|{\xrightarrow {\mathrm {a.s.} }}\ 0.}
This result is useful to derive consistency of a large class of estimators (see Extremum estimator).
Borel's law of large numbers[edit]
Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event occurs approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if E denotes the event in question, p its probability of occurrence, and Nn(E) the number of times E occurs in the first n trials, then with probability one,[21]
N n ( E ) n → p as n → ∞ . {\displaystyle {\frac {N_{n}(E)}{n}}\to p{\text{ as }}n\to \infty .}
This theorem makes rigorous the intuitive notion of probability as the long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory.
Chebyshev's inequality. Let X be a random variable with finite expected value μ and finite non-zero variance σ2. Then for any real number k > 0,
Pr ( | X − μ | ≥ k σ ) ≤ 1 k 2 . {\displaystyle \Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.}
Proof of the weak law[edit]
Given X1, X2, ... an infinite sequence of i.i.d. random variables with finite expected value E(X1) = E(X2) = ... = µ < ∞, we are interested in the convergence of the sample average
X ¯ n = 1 n ( X 1 + ⋯ + X n ) . {\displaystyle {\overline {X}}_{n}={\tfrac {1}{n}}(X_{1}+\cdots +X_{n}).}
The weak law of large numbers states:
Theorem: X ¯ n → P μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 2)
Proof using Chebyshev's inequality assuming finite variance[edit]
This proof uses the assumption of finite variance Var  ( X i ) = σ 2 {\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} (for all i {\displaystyle i} ). The independence of the random variables implies no correlation between them, and we have that
Var  ( X ¯ n ) = Var  ( 1 n ( X 1 + ⋯ + X n ) ) = 1 n 2 Var  ( X 1 + ⋯ + X n ) = n σ 2 n 2 = σ 2 n . {\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}
The common mean μ of the sequence is the mean of the sample average:
E ( X ¯ n ) = μ . {\displaystyle E({\overline {X}}_{n})=\mu .}
Using Chebyshev's inequality on X ¯ n {\displaystyle {\overline {X}}_{n}} results in
P  ( | X ¯ n − μ | ≥ ε ) ≤ σ 2 n ε 2 . {\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\leq {\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}
This may be used to obtain the following:
P  ( | X ¯ n − μ | < ε ) = 1 − P  ( | X ¯ n − μ | ≥ ε ) ≥ 1 − σ 2 n ε 2 . {\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|<\varepsilon )=1-\operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\geq 1-{\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}
As n approaches infinity, the expression approaches 1. And by definition of convergence in probability, we have obtained
X ¯ n → P μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 2)
Proof using convergence of characteristic functions[edit]
By Taylor's theorem for complex functions, the characteristic function of any random variable, X, with finite mean μ, can be written as
φ X ( t ) = 1 + i t μ + o ( t ) , t → 0. {\displaystyle \varphi _{X}(t)=1+it\mu +o(t),\quad t\rightarrow 0.}
All X1, X2, ... have the same characteristic function, so we will simply denote this φX.
Among the basic properties of characteristic functions there are
φ 1 n X ( t ) = φ X ( t n ) and φ X + Y ( t ) = φ X ( t ) φ Y ( t ) {\displaystyle \varphi _{{\frac {1}{n}}X}(t)=\varphi _{X}({\tfrac {t}{n}})\quad {\text{and}}\quad \varphi _{X+Y}(t)=\varphi _{X}(t)\varphi _{Y}(t)\quad } if X and Y are independent.
These rules can be used to calculate the characteristic function of X ¯ n {\displaystyle \scriptstyle {\overline {X}}_{n}} in terms of φX:
φ X ¯ n ( t ) = [ φ X ( t n ) ] n = [ 1 + i μ t n + o ( t n ) ] n → e i t μ , as n → ∞ . {\displaystyle \varphi _{{\overline {X}}_{n}}(t)=\left[\varphi _{X}\left({t \over n}\right)\right]^{n}=\left[1+i\mu {t \over n}+o\left({t \over n}\right)\right]^{n}\,\rightarrow \,e^{it\mu },\quad {\text{as}}\quad n\rightarrow \infty .}
The limit eitμ is the characteristic function of the constant random variable μ, and hence by the Lévy continuity theorem, X ¯ n {\displaystyle \scriptstyle {\overline {X}}_{n}} converges in distribution to μ:
X ¯ n → D μ for n → ∞ . {\displaystyle {\overline {X}}_{n}\,{\xrightarrow {\mathcal {D}}}\,\mu \qquad {\text{for}}\qquad n\to \infty .}
μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see Convergence of random variables.) Therefore,
X ¯ n → P μ when n → ∞ . {\displaystyle {\begin{matrix}{}\\{\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \qquad {\textrm {when}}\ n\to \infty .\\{}\end{matrix}}}
(law. 2)
This shows that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists.
See also[edit]
Asymptotic equipartition property
Central limit theorem
Infinite monkey theorem
Law of averages
Law of the iterated logarithm
Lindy effect
Regression toward the mean
Sortition
Notes[edit]
Jump up ^ Mlodinow, L. The Drunkard's Walk. New York: Random House, 2008. p. 50.
Jump up ^ Jakob Bernoulli, Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis, 1713, Chapter 4, (Translated into English by Oscar Sheynin)
Jump up ^ Poisson names the "law of large numbers" (la loi des grands nombres) in: S.D. Poisson, Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés (Paris, France: Bachelier, 1837), p. 7. He attempts a two-part proof of the law on pp. 139–143 and pp. 277 ff.
Jump up ^ Hacking, Ian. (1983) "19th-century Cracks in the Concept of Determinism", Journal of the History of Ideas, 44 (3), 455-475 JSTOR 2709176
Jump up ^ Tchebichef, P. (1846). "Démonstration élémentaire d'une proposition générale de la théorie des probabilités". Journal für die reine und angewandte Mathematik (Crelles Journal). 1846 (33): 259–267. doi:10.1515/crll.1846.33.259.
^ Jump up to: a b Seneta 2013.
^ Jump up to: a b Yuri Prohorov. "Law of large numbers". Encyclopedia of Mathematics.
Jump up ^ Etemadi, N.Z. (1981). "An elementary proof of the strong law of large numbers". Wahrscheinlichkeitstheorie verw Gebiete. 55 (1): 119–122. doi:10.1007/BF01013465.
Jump up ^ Loève 1977, Chapter 1.4, p. 14
Jump up ^ Loève 1977, Chapter 17.3, p. 251
Jump up ^ "The strong law of large numbers – What's new". Terrytao.wordpress.com. Retrieved 2012-06-09.
^ Jump up to: a b Yuri Prokhorov. "Strong law of large numbers". Encyclopedia of Mathematics.
Jump up ^ Ross (2009)
Jump up ^ Lehmann, Erich L; Romano, Joseph P (2006-03-30). Weak law converges to constant. ISBN 9780387276052.
Jump up ^ "A NOTE ON THE WEAK LAW OF LARGE NUMBERS FOR EXCHANGEABLE RANDOM VARIABLES" (PDF). Dguvl Hun Hong and Sung Ho Lee.
Jump up ^ "weak law of large numbers: proof using characteristic functions vs proof using truncation VARIABLES".
Jump up ^ Mukherjee, Sayan. "Law of large numbers" (PDF). Archived from the original (PDF) on 2013-03-09. Retrieved 2014-06-28.
Jump up ^ J. Geyer, Charles. "Law of large numbers" (PDF).
Jump up ^ Newey & McFadden 1994, Lemma 2.4
Jump up ^ Jennrich, Robert I. (1969). "Asymptotic Properties of Non-Linear Least Squares Estimators". The Annals of Mathematical Statistics. 40 (2): 633–643. doi:10.1214/aoms/1177697731.
Jump up ^ An Analytic Technique to Prove Borel's Strong Law of Large Numbers Wen, L. Am Math Month 1991
References[edit]
Grimmett, G. R.; Stirzaker, D. R. (1992). Probability and Random Processes, 2nd Edition. Clarendon Press, Oxford. ISBN 0-19-853665-8.
Richard Durrett (1995). Probability: Theory and Examples, 2nd Edition. Duxbury Press.
Martin Jacobsen (1992). Videregående Sandsynlighedsregning (Advanced Probability Theory) 3rd Edition. HCØ-tryk, Copenhagen. ISBN 87-91180-71-6.
Loève, Michel (1977). Probability theory 1 (4th ed.). Springer Verlag.
Newey, Whitney K.; McFadden, Daniel (1994). Large sample estimation and hypothesis testing. Handbook of econometrics, vol. IV, Ch. 36. Elsevier Science. pp. 2111–2245.
Ross, Sheldon (2009). A first course in probability (8th ed.). Prentice Hall press. ISBN 978-0-13-603313-4.
Sen, P. K; Singer, J. M. (1993). Large sample methods in statistics. Chapman & Hall, Inc.
Seneta, Eugene (2013), "A Tricentenary history of the Law of Large Numbers", Bernoulli, 19 (4): 1088–1121, arXiv:1309.6488, doi:10.3150/12-BEJSP12
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Law of large numbers", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Weak Law of Large Numbers". MathWorld.
Weisstein, Eric W. "Strong Law of Large Numbers". MathWorld.
Animations for the Law of Large Numbers by Yihui Xie using the R package animation
Apple CEO Tim Cook said something that would make statisticians cringe. "We don't believe in such laws as laws of large numbers. This is sort of, uh, old dogma, I think, that was cooked up by somebody [..]" said Tim Cook and while: "However, the law of large numbers has nothing to do with large companies, large revenues, or large growth rates. The law of large numbers is a fundamental concept in probability theory and statistics, tying together theoretical probabilities that we can calculate to the actual outcomes of experiments that we empirically perform. explained Business Insider
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Cyclone Kathy
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Severe Tropical Cyclone Kathy
Category 5 severe tropical cyclone (Aus scale)
Category 4 (Saffir–Simpson scale)
Cyclone Kathy near peak intensity on 22 March over the Gulf of Carpentaria
Formed
16 March 1984
Dissipated
23 March 1984
Highest winds
10-minute sustained: 205 km/h (125 mph)
1-minute sustained: 250 km/h (155 mph)
Highest gust
Gusts: 280 km/h (175 mph)
Lowest pressure
916 hPa (mbar); 27.05 inHg
Fatalities
1 direct
Damage
$11.4 million (1984 USD)
Areas affected
Far North Queensland and Northern Territory
Part of the 1983–84 Australian region cyclone season
Severe Tropical Cyclone Kathy was a powerful tropical cyclone that devastated the Sir Edward Pellew Group of Islands in March 1984. Originating from a tropical low off the southern coast of Papua New Guinea. Tracking westward, the system attained gale-force winds by 18 March before striking the Cape York Peninsula. After crossing the area, Kathy entered the Gulf of Carpentaria where environmental conditions favoured significant development. On 22 March, the storm attained its peak intensity as a Category 5 severe tropical cyclone (Australian tropical cyclone intensity scale) with ten-minute sustained winds of 205 km/h (125 mph). By this time, the storm had been tracking towards the southwest and struck the Sir Edward Pellew Group of Islands later on 22 March before moving over the Australian mainland as a slightly weaker system. Once over land, Kathy rapidly degraded, losing gale-force winds within 24 hours; the storm dissipated over the Northern Territory on 24 March.
Moving through the Pellew Islands as a powerful cyclone, Kathy generated a maximum storm surge of 4.2 m (14 ft) and winds exceeding 232 km/h (144 mph). Offshore, 20 vessels were affected, one of which sank and resulted in the death of its captain. In mainland Australia, the system struck a relatively sparsely populated area, with only Borroloola being significantly impacted by the cyclone. Additionally, mangroves at along the McArthur River sustained substantial damage. Overall, losses from the storm were placed at A$12 million (A$30.5 million 2011 AUD; US$24.1 million 2011 USD). Following its usage, the name Kathy was retired and will never be used again to name an Australian cyclone.
Contents
[hide]
1 Meteorological history
2 Impact
3 See also
4 References
5 External links
Meteorological history[edit]
Map plotting the track and intensity of the storm, according to the Saffir–Simpson scale
On 16 March 1984,[1] the Bureau of Meteorology began monitoring a tropical low off the southern coast of Papua New Guinea for possible development into a tropical cyclone. Over the following day, the system quickly developed and attained gale-force winds as it neared the Cape York Peninsula. Attaining a pressure of 990 hPa (mbar; 29.23 inHg), the weak system made landfall just north of Weipa, Queensland on 19 March before losing strength.[2] Though already considered a tropical cyclone for two days by the Bureau, the Joint Typhoon Warning Center (JTWC) issued their first advisory on the system just prior to its landfall; however, they considered it to be a tropical depression rather than a storm.[3] After roughly 12 hours over land, a weakened Kathy emerged over the warm waters of the Gulf of Carpentaria on 20 March.[2] Once over water, the JTWC classified the system as a tropical storm.[3] Turning southwestward, the storm entered a region favoring significant development and soon began a period of rapid intensification.[2]
Intensification took place through 22 March as convection deepened around Kathy's centre and a well-defined eye formed.[2] The system reached its peak strength as a Category 5 severe tropical cyclone on the Australian tropical cyclone intensity scale with ten-minute sustained winds of 205 km/h (125 mph). Operational analysis of the storm indicated a minimum pressure of 920 hPa (mbar; 27.17 inHg);[1] however, a reassessment in 2009 concluded that Kathy's pressure had been slightly lower, bottoming out at 916 hPa (mbar; 27.05 inHg). Additionally, the JTWC assessed the system to have been slightly stronger, estimating peak one-minute sustained winds at 250 km/h (155 mph).[4] At this time, the storm was estimated to have a Dvorak technique rating of 7.0, equivalent to a Category 5 on the Saffir–Simpson Hurricane Scale;[5]
As Cyclone Kathy neared the Australia coastline later on 22 March, it began to weaken as it interacted with land.[2] Radar imagery depicted the system as having an eye 10.8 km (6.7 mi) in diameter.[5] Roughly 5 hours after peaking, the storm passed directly over the Sir Edward Pellew Group of Islands, resulting in a degrading structure.[2][5] Near midnight, the system weakened to a Category 4 cyclone and made landfall in a remote region near Borroloola.[6] Based on a wind gust to sustained ratio of 1.4, Kathy was estimated to have moved onshore with winds between 165 and 185 km/h (105 and 115 mph).[5][7] In the hours after landfall, Kathy rapid weakened, no longer retaining gale-force winds once 200 km (120 mi) away from the Gulf on 23 March. Later that day, the system turned south-westward and persisted for another 24 hours before dissipating.[2]
Impact[edit]
In Far North Queensland, Kathy brought moderate rainfall and strong winds to much of the region. No measurements from the centre of the storm were made, although a wind gust of 60 km/h (37 mph) was recorded in Batavia Downs, Queensland, about 55 km (34 mi) south-southeast of the cyclone.[2] Between 18 and 21 March, a total of 126.7 mm (4.99 in) of rain fell near Batavia Downs.[8] Due to the relatively low intensity of the storm, damage was minimal and mainly confined to trees.[2]
Striking the Sir Edward Pellew Group of Islands as a high-end Category 4 cyclone, Kathy wrought significant damage in the region. A storm surge of 4.2 m (14 ft) struck Vanderlin Island, destroying much of the nesting ground for sea turtles. Winds were recorded up to 232 km/h (144 mph) before the station's anemometer failed. Several camps across the islands were destroyed.[2] Offshore, 20 vessels were caught in the storm, one of which sank and three others ran aground.[9] One person perished on board the ship that sank during the storm. Losses sustained by the ships were placed at A$5 million (A$12.7 million 2011 AUD;[nb 1] US$10 million 2011 USD[nb 2]).[9]
Though a powerful storm, only light to moderate rains fell along the cyclone's path with totals generally less than 100 mm (3.9 in). The remnants, however, triggered thunderstorms near Tennant Creek that produced 325 mm (12.8 in) of rain in 24 hours. Reports indicate that several sea turtles and dugongs were washed up to 8 km (5.0 mi) inland by Kathy's surge. The only population center significantly affected by the storm was Borroloola, where moderate to severe damage took place.[9] At least one home was destroyed and many more were damaged in the town.[13] Along the mouth of the McArthur River, near where Kathy struck the mainland, the combination of severe winds and storm surge killed 80% (±20%) of the Rhizophoraceae mangrove trees in the region. Along the entire length of the river, 45% (±15.3%) of the Rhizophoraceae, Bruguiera and Ceriops as well as 18.5% (±11.4%) of the Exoecaria agallocha, Lumnitzera racemosa and Avicennia marina perished.[7] In all, losses from the storm were placed at A$12 ,million (A$30.5 million 2011 AUD;[nb 1] US$24.1 million 2011 USD[nb 2]).[14]
In the wake of the storm, Australian Prime Minister Bob Hawke promised residents of Borroloola that they would receive all possible help from the government. About 400 people in the town were left homeless by Cyclone Kathy. By 25 March, a Royal Australian Air Force aircraft was sent to the region carrying emergency supplies for sheltering displaced persons. Two patrol boats from the Royal Australian Navy were also sent to assist in relief efforts. One brought fuel and medical supplies and the other assisted trawlers set adrift or grounded by the storm.[15] Following its usage, the name Kathy was retired from Brisbane's list of cyclone names and will never be used again.[16][citation needed]
See also[edit]
Tropical cyclones portal
Lists of tropical cyclones
List of retired Australian cyclone names
References[edit]
^ Jump up to: a b "Raw cyclone track data" (XLS). Australian Bureau of Meteorology. 2007. Retrieved 24 February 2011.
^ Jump up to: a b c d e f g h i j "Severe Tropical Cyclone Kathy". Bureau of Meteorology. 2011. Retrieved 24 February 2011.
^ Jump up to: a b Joint Typhoon Warning Center (1985). "Tropical Cyclone 30P Best Track" (TXT). United States Navy. Retrieved 24 February 2011.
Jump up ^ Neumann (22 May 2009). "Southern Hemisphere Best Tracks". National Oceanic and Atmospheric Administration. Retrieved 24 February 2011.
^ Jump up to: a b c d Bruce Harper (2002). "Tropical Cyclone Parameter Estimation in the Australian Region: Wind-Pressure Relationships and Related Issues for Engineering Planning and Design" (PDF). Woodside Energy Ltd. pp. 73–74. Retrieved 24 February 2011.
Jump up ^ "Tropical Cyclone Track Maps: Kathy 1984". Australia Severe Weather. 11 August 2009. Retrieved 24 February 2011.
^ Jump up to: a b T. J. Smith III (1986). "Comparative Effects of Cyclone Damage to Mangrove Forests: Kathy Versus Winifred" (PDF). Australian Institute of Marine Science. Retrieved 25 February 2011.
Jump up ^ "1984 Daily Rainfall Summary for Moreton Telegraph Station, Queensland". Bureau of Meteorology. 1984. Retrieved 23 February 2011.
^ Jump up to: a b c "Tropical Cyclone Kathy: Northern Territory". Bureau of Meteorology. 2011. Retrieved 24 February 2011.
Jump up ^ "Inflation Calculator". Reserve Bank of Australia. 2010. Retrieved 29 January 2011.
Jump up ^ "US Dollar Per Australian Dollar". Data 360. 2011. Retrieved 29 January 2011.
Jump up ^ "What is a dollar worth?". Federal Reserve Bank of Minneapolis. 2018. Retrieved 2018. Check date values in: |access-date= (help)
Jump up ^ Staff Writer (6 February 2011). "Found at last: the drover's wife". The Sydney Morning Herald. Archived from the original on February 25, 2011. Retrieved 25 February 2011.
Jump up ^ "Event — Gulf of Carpentaria, Qld & NT: Cyclone (incl storm surge)". Government of Australia. 2011. Archived from the original on 6 July 2011. Retrieved 25 February 2011.
Jump up ^ Staff Writer (25 March 1984). "Cyclone Kathy Lashes North". The Sydney Morning Herald. p. 8. Retrieved 25 February 2011.
Jump up ^ "Tropical Cyclone Names". Bureau of Meteorology. 2010.
Notes
^ Jump up to: a b Australian dollar inflation rates obtained through the consumer price index.[10]
^ Jump up to: a b United States dollar values obtained through historic conversion rates and the consumer price index.[11][12]
External links[edit]
Joint Typhoon Warning Center (JTWC).
Australian Bureau of Meteorology.
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Tom vs Time
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Tom vs Time
Genre
Reality
Docuseries
Created by
Gotham Chopra
Directed by
Gotham Chopra
Starring
Tom Brady
Gisele Bündchen
Composer(s)
Antonio Pinto
Ed Côrtes
Country of origin
United States
Original language(s)
English
No. of seasons
1
No. of episodes
5 (list of episodes)
Production
Executive producer(s)
Gotham Chopra
Ameeth Sankaran
Chris Uettwiller
Martin Desmond Roe
Producer(s)
Nicolas Emiliani
Erik LeDrew
Editor(s)
Brady Hammes
Running time
15–21 minutes
Production company(s)
Dirty Robber
Religion of Sports
Release
Original network
Facebook Watch
Original release
January 25, 2018 (2018-01-25) – present
External links
Official website
Tom vs Time is an American reality web television series that airs on Facebook Watch. It follows New England Patriots quarterback Tom Brady and provides an intimate look at his off-season training regimen as well as his home life spent with wife Gisele Bündchen and their children. The first season is set to consist of six episodes and premiered on January 25, 2018.
The series was created by Gotham Chopra who directs and executive produces the show as well.
Contents
[hide]
1 Premise
2 Production
2.1 Development
2.2 Marketing
3 Episodes
4 See also
5 References
6 External links
Premise[edit]
Tom vs Time was filmed during "the Patriots’ winning 2016 season (Brady’s fifth Super Bowl win)" and provides "an intimate look at Brady’s home life, including his two children with wife Gisele Bündchen."[1]
Production[edit]
Development[edit]
On January 9, 2018, it was announced that Facebook Watch had ordered a first season of Tom vs Time, a new docuseries starring football quarterback Tom Brady. The series was created by Gotham Chopra who previously worked with Brady on Audience Network's Religion of Sports series. Brady granted Chopra extensive access to his life. It was announced that episodes are set to include footage shot at Brady's Brookline, Massachusetts home, with his children and wife, Gisele Bündchen, on family retreats to Costa Rica, on a trip to Montana (with teammates Julian Edelman and Danny Amendola), on a summer tour of China with his son Jack, and in the car during Brady’s commute to and from work in Foxborough, Massachusetts.[2][3][4]
Marketing[edit]
Simultaneously with the initial series announcement, Facebook released a trailer for the first season of the show.[5][6][7][8]
Episodes[edit]
No.
Title
Original release date
1
"The Physical Game" January 25, 2018 (2018-01-25)
Following the Patriots' victory at Super Bowl LI, Tom Brady looks towards the next season and demonstrates the unique and well honed training process he undergoes during the offseason.
2
"The Mental Game" January 28, 2018 (2018-01-28)
Brady explains how football is just as much a mental game for him as it is a physical one. And despite all of his offseason preparation, the Patriots narrowly lose their fourth game of the 2017 season. But Brady explains how having his family at home waiting for him and having another game the following week does not allow him time to dwell on a loss.
3
"The Social Game" January 30, 2018 (2018-01-30)
Brady takes some of his closest teammates to Yellowstone National Park for a brotherhood bonding experience aka “Gladiator Camp.” Even on these vacations, he and his teammates find themselves continuing to train.
4
"The Emotional Game" February 1, 2018 (2018-02-01)
5
"The Spiritual Game" February 4, 2018 (2018-02-04)
See also[edit]
List of original programs distributed by Facebook Watch
References[edit]
Jump up ^ Carr, Mary Kate (January 10, 2018). "Tom Brady has a new docuseries, 'Tom vs. Time,' and here's the trailer". Entertainment Weekly. Retrieved January 27, 2018.
Jump up ^ Leibovich, Mark (January 9, 2018). "Tom Brady Gave a Filmmaker Unusual Access to His Private Life". The New York Times. Retrieved January 27, 2018.
Jump up ^ Emery, Debbie (January 9, 2018). "Tom Brady Details Battle With Age in 'Tom vs Time' Docu-Series (Video)". The Wrap. Retrieved January 27, 2018.
Jump up ^ Grossman, Evan (January 9, 2018). "Tom Brady to reveal private life in doc series 'Tom vs. Time'". NY Daily News. Retrieved January 27, 2018.
Jump up ^ Yang, Nicole (January 9, 2018). "Tom Brady is releasing a documentary series called 'Tom vs. Time'". Boston.com. Retrieved January 27, 2018.
Jump up ^ Benjamin, Cody (January 9, 2018). "WATCH: Tom Brady stars in upcoming documentary series 'Tom vs. Time'". CBS Sports. Retrieved January 27, 2018.
Jump up ^ Carroll, Charlotte (January 9, 2018). "Watch: Tom Brady documentary trailer released". Sports Illustrated. Retrieved January 27, 2018.
Jump up ^ Fiske, Angelique (January 9, 2018). "Watch first trailer for Brady doc, 'Tom vs Time'". New England Patriots. Retrieved January 27, 2018.
External links[edit]
Official website
Tom vs Time on IMDb
[hide]
v
t
e
Facebook Watch original programming
Current
Ball in the Family (since 2017)
Humans of New York: The Series (since 2017)
I Want My Phone Back (since 2017)
Returning the Favor (since 2017)
Strangers (since 2017)
RelationShipped (since 2017)
The Mind of a Chef (since 2017)
Bill Murray & Brian Doyle-Murray's Extra Innings (since 2017)
Loosely Exactly Nicole (since 2017)
WWE Mixed Match Challenge (since 2018)
Tom vs Time (since 2018)
Upcoming
Shame (2018)
Five Points (TBA)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Tom_vs_Time&oldid=827777358"
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This page was last edited on 26 February 2018, at 17:55.
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