MATH 367, spg 98, CHAPTER 5 Section 1, Mar 9 Chebishev’s Inequality. Law of Large Numbers.

Definition: The standard deviation of a random variable X is σ = \sqrt{Var(X)}.

Theorem: Let X be a random variable with mean m and variance σ^{2}. Then

P (|X − m| > t) ≤ \frac{σ^{2}}{t^{2}}

Letting t = cσ,

P (|X − m| > cσ) ≤ \frac{1}{c^{2}}

Other incarnations of this theorem are: Theorem: Let Y be a Bernoulli random variable B(n, p). Then

P (|Y/n − p| > t) ≤ \frac{1}{nt^{2}} .

In particular, given any number s > 0, there exists an N such that for all n ≥ N, if Y is B(n, p) then

P (|Y/n − p| > t) < s.

Theorem: Let X_{1}, X_{2}, …, X_{n} be independent random variables all with the same mean m and variance σ^{2}. Then

P (|\frac{X_{1} + … + X_{n}}{n} − m| ≥ t) ≤ \frac{σ^{2}}{nt^{2}} .

In particular, given any number s > 0, there exists an N such that for all n ≥ N,

P (|\frac{X_{1} + … + X_{n}}{n} − m| ≥ t) ≤ s.

For next class read section 5.1 and do problems 4–7, page 122.