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Convergence in Distribution

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- 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.

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- 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)

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- 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…

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- 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.

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- 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).

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- Suppose X is Poisson(900) random variable. Find P(X > 950).

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- 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.

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- 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).

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- 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 ∞.

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- 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?

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