Question

give an example of a discrete random variable X whose values are integers and such that E(X) = infinite. Prove that E(X) = infinite for your example. (hints: if you will be paid 2^k dollars for the kth head when you flip a fair coin., the expected value is infinite...) Or give other easy examples.

Answer 1

Example of a Random Variable X with the properties:

(i) X is a Discrete Variable

(ii) X takes integers

(iii) E(X) = infinity.

EXAMPLE OF A VARIABLE WITH ALL THE 3 PROPERTIES:

,

for x = 1,2,3,...

This satisfies all the 3 conditions:

(i) X is a discrete variable, since X takes values 1,2,3,..only

(ii) Values of X are integers:X = 1,2,3,.

(iii)

By Theorem:

is a divergent sries.

So,

we nore:
E(X) = .

Thus,all the 3 properties are satisfied.

P(X) = r(r + 1)

CONCTRETE EXAMPLE:

Consider the following game:

(i) We flip a coin until it lands tails.

(ii) We win 2ⁿ dollars,

where

n is the number of heads we get.

Let

X = Amount of money we get.

In this game,

The Expected Value of X is given by: