1 / 14

Law of Large Numbers

Law of Large Numbers. Toss a coin n times. Suppose X i ’s are Bernoulli random variables with p = ½ and E ( X i ) = ½. The proportion of heads is . Intuitively approaches ½ as n  ∞. Markov’s Inequality.

Download Presentation

Law of Large Numbers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Law of Large Numbers • Toss a coin n times. • Suppose • Xi’s are Bernoulli random variables with p = ½ and E(Xi) = ½. • The proportion of heads is . • Intuitively approaches ½ as n  ∞. week 12

  2. Markov’s Inequality • If X is a non-negative random variable with E(X) < ∞ and a >0 then, week 12

  3. Chebyshev’s Inequality • For a random variable X with E(X) < ∞ and V(X) < ∞, for any a >0 • Proof: week 12

  4. Back to the Law of Large Numbers • Interested in sequence of random variables X1, X2, X3,… such that the random variables are independent and identically distributed (i.i.d). Let Suppose E(Xi) = μ , V(Xi) = σ2, then and • Intuitively, as n  ∞, so week 12

  5. Formally, the Weak Law of Large Numbers (WLLN) states the following: • Suppose X1, X2, X3,…are i.i.d with E(Xi) = μ < ∞ , V(Xi) = σ2 < ∞, then for any positive number a as n ∞ . This is called Convergence in Probability. Proof: week 12

  6. Example • Flip a coin 10,000 times. Let • E(Xi) = ½ and V(Xi) = ¼ . • Take a = 0.01, then by Chebyshev’s Inequality • Chebyshev Inequality gives a very weak upper bound. • Chebyshev Inequality works regardless of the distribution of the Xi’s. week 12

  7. Strong Law of Large Number • Suppose X1, X2, X3,…are i.i.d with E(Xi) = μ < ∞ , then converges to μ as n  ∞ with probability 1. That is • This is called convergence almost surely. week 12

  8. Continuity Theorem for MGFs • Let X be a random variable such that for some t0 > 0 we have mX(t) < ∞ for . Further, if X1, X2,…is a sequence of random variables with and for all then {Xn} converges in distribution to X. • This theorem can also be stated as follows: Let Fn be a sequence of cdfs with corresponding mgf mn. Let F be a cdf with mgf m. If mn(t) m(t) for all t in an open interval containing zero, then Fn(x) F(x) at all continuity points of F. • Example: Poisson distribution can be approximated by a Normal distribution for large λ. week 12

  9. Example to illustrate the Continuity Theorem • Let λ1, λ2,…be an increasing sequence with λn∞ as n ∞ and let {Xi} be a sequence of Poisson random variables with the corresponding parameters. We know that E(Xn) = λn = V(Xn). • Let then we have that E(Zn) = 0, V(Zn) = 1. • We can show that the mgf of Zn is the mgf of a Standard Normal random variable. • We say that Zn convergence in distribution to Z ~ N(0,1). week 12

  10. Example • Suppose X is Poisson(900) random variable. Find P(X > 950). week 12

  11. Central Limit Theorem • The central limit theorem is concerned with the limiting property of sums of random variables. • If X1, X2,…is a sequence of i.i.d random variables with mean μ and variance σ2 and , then by the WLLN we have that in probability. • The CLT concerned not just with the fact of convergence but how Sn/n fluctuates around μ. • Note that E(Sn) = nμ and V(Sn) = nσ2. The standardized version of Sn is and we have that E(Zn) = 0, V(Zn) = 1. week 12

  12. The Central Limit Theorem • Let X1, X2,…be a sequence of i.i.d random variables with E(Xi) = μ < ∞ and Var(Xi) = σ2 < ∞. Suppose the common distribution function FX(x) and the common moment generating function mX(t) are defined in a neighborhood of 0. Let Then, for - ∞ < x < ∞ where Ф(x) is the cdf for the standard normal distribution. • This is equivalent to saying that converges in distribution to Z ~ N(0,1). • Also, i.e. converges in distribution to Z ~ N(0,1). week 12

  13. Example • Suppose X1, X2,…are i.i.d random variables and each has the Poisson(3) distribution. So E(Xi) = V(Xi) = 3. • The CLT says that as n  ∞. week 12

  14. Examples • A very common application of the CLT is the Normal approximation to the Binomial distribution. • Suppose X1, X2,…are i.i.d random variables and each has the Bernoulli(p) distribution. So E(Xi) = p and V(Xi) = p(1- p). • The CLT says that as n  ∞. • Let Yn = X1 + … + Xn then Yn has a Binomial(n, p) distribution. So for large n, • Suppose we flip a biased coin 1000 times and the probability of heads on any one toss is 0.6. Find the probability of getting at least 550 heads. • Suppose we toss a coin 100 times and observed 60 heads. Is the coin fair? week 12

More Related