Grinstead-Snell-Probability · Problem 26

Expected Value and Variance

Mathematics · variance-discrete-random-variables

Problem

For a sequence of Bernoulli trials, let

X_1

be the number of trials until the first success. For

j \geq 2

, let

X_j

be the number of trials after the

(j-1)

st success until the

j

th success. It can be shown that

X_1, X_2, \ldots

is an independent trials process. ((a)) What is the common distribution, expected value, and variance for

X_j

? ((b)) Let

T_n = X_1 + X_2 + \cdots + X_n

. Then

T_n

is the time until the

n

th success. Find

E(T_n)

and

V(T_n)

. ((c)) Use the results of (b) to find the expected value and variance for the number of tosses of a coin until the

n

th occurrence of a head.