Introduction to Probability · Problem 8.2.13

Central Limit Theorem — Grinstead & Snell

Mathematics — Probability · law-of-large-numbers-continuous

Problem

Suppose one hundred numbers

X_1, X_2, \ldots, X_{100}

are chosen independently at random from

[0, 20]

. Let

S = X_1 + X_2 + \cdots + X_{100}

be the sum,

A = S/100

the average, and

S^* = (S - 1000)/(10/\sqrt{3})

the standardized sum. Find lower bounds for the probabilities (a)

P(|S - 1000| \leq 100)

. (b)

P(|A - 10| \leq 1)

. (c)

P(|S^*| \leq \sqrt{3})