Convergence in Distribution

1. Convergence in Distribution • Recall: in probability if • Definition Let X1, X2,…be a sequence of random variables with cumulative distribution functions F1, F2,… and let X be a random variable with cdf FX(x). We say that the sequence {Xn} converges in distribution to X if at every point x in which F is continuous. • This can also be stated as: {Xn} converges in distribution to X if for all such that P(X = x) = 0 • Convergence in distribution is also called “weak convergence”. It is weaker then convergence in probability. We can show that convergence in probability implies convergence in distribution. week 11

2. Simple Example • Assume n is a positive integer. Further, suppose that the probability mass function of Xn is: Note that this is a valid p.m.f for n ≥ 2. • For n ≥ 2, {Xn} convergence in distribution to X which has p.m.f P(X = 0) = P(X = 1) = ½ i.e. X ~ Bernoulli(1/2) week 11

3. Example • X1, X2,…is a sequence of i.i.d random variables with E(Xi) = μ < ∞. • Let . Then, by the WLLN for any a > 0 as n  ∞. • So… week 11

4. 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 11

5. 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 11

6. Example • Suppose X is Poisson(900) random variable. Find P(X > 950). week 11

7. 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 11

8. 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 11

9. 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 11

10. 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 11