Convergence in distribution
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Convergence in Distribution. Recall: in probability if Definition Let X 1 , X 2 ,…be a sequence of random variables with cumulative distribution functions F 1 , F 2 ,… and let X be a random variable with cdf

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Convergence in Distribution

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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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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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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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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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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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Example

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

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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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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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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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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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