Class Notes in Statistics and Econometrics Part 4 pps

be a sequence of independent random variables all of whichhave the same expected value µ and variance σ2.. The empirical distribution is a discrete probability distribution defined by Pr

Chebyshev Inequality, Weak Law of Large

Numbers, and Central Limit Theorem

to.”) One does not need to know the full distribution ofyfor that, only its expectedvalue and standard deviation We will give here a proof only if y has a discretedistribution, but the inequality is valid in general Going over to the standardizedvariable z = y−µσ we have to show Pr[|z|≥k] ≤ 1

k 2 Assuming z assumes the values

z1, z2, with probabilities p(z1), p(z2), , then

i : |z i|≥k

zi2p(zi)(7.1.4)

2k 2, and 0 with probability 1 − k12, has expected value 0 and variance 1 and the

≤-sign in (7.1.1) becomes an equal sign

Problem 115 [HT83, p 316] Lety be the number of successes in n trials of aBernoulli experiment with success probability p Show that

n− p <ε≥ 1 − 1

4nε2.Hint: first compute what Chebyshev will tell you about the lefthand side, and thenyou will need still another inequality

Answer E[ y /n] = p and var[ y /n] = pq/n (where q = 1 − p) Chebyshev says therefore

 y

n− p

≥ k

qpq n



≤ 1

k 2 Setting ε = kppq/n, therefore 1/k 2 = pq/nε 2 one can rewerite ( 7.1.7 ) as

 y

n− p

... written as an integral (i.e.,

an infinite sum each summand of which is infinitesimal), therefore we get