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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

Chebyshev’s Inequality

- For a random variable X with E(X) < ∞ and V(X) < ∞, for any a >0
- Proof:

week 12

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

- 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

Example following:

- 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

Strong Law of Large Number following:

- 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

Continuity Theorem for MGFs following:

- 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

Example to illustrate the Continuity Theorem following:

- 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

Central Limit Theorem following:

- 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

The Central Limit Theorem following:

- 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

Example following:

- 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

Examples following:

- 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

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