Chapter 6: Large Random Samples

6.2 The Law of Large Numbers

Theorem: Chebyshev Inequality. Let X be a random

variable for which Var(X) exists. Then for every number

t > 0,

P (|X E(X)| t) V ar(X)t2

Theorem: Let X1, ..., Xn be a random sample from a

distribution with mean and variance 2. Let Xn be the

sample mean. Then

E(X) = , V ar(X) =2

n.

P (|X | t) 2

nt2

Example: Suppose that a random sample is to be taken

from a distribution for which the value of the mean is

not known, but the standard deviation 2 units. Weshall determine how large the sample size such that

P (|X | < 1) 0.99

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From Chebyshev Inequality,

P (|X | 1) 2

n 4n

it follows that n must be chosen so that 4n 0.01, n 400.

Definition: Convergence in Probability. A sequence

Z1, Z2, ... of random variables converges to b in probabil-

ity if for every number > 0,

limn

P (|Zn n| < ) = 1

This property is denoted by Znp b.

Theorem: Law of Large Numbers. Suppose thatX1, ..., Xn

form a random sample from a distribution for which the

mean is and for which the variance is finite. Let Xn

denote the sample mean. Then

Xnp

Theorem: If Znp b, and if g(z) is a function that is

continuous at z = b, then g(Zn)p g(b).

6.3 The Central Limit Theorem

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Theorem: Central Limit Theorem . If the random

variables X1, ..., Xn form a random sample of size n from

a given distribution with mean and variance 2 (0