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.
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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)
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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…
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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.
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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).
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6. Example Suppose X is Poisson(900) random variable. Find P(X > 950).
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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.
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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).
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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  ∞.
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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?
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