Question

Let X have the probability mass function P(X = −1) = 1 2 , P(X = 0) = 1 3 , P(X = 1) = 1 6 Calculate E(|X|) using the approaches in (a) and (b) below. (a) First find the probability mass function of the random variable Y = |X| and using that compute E(|X|). (b) Apply formula (3.24) with g(x) = |x|. For reference, formula 3.24 states

Answer 1

Explanation:

From the given information:

Note that the possible values of Y are 0 and 1 because;

y = 0 if X = 0 and y = 1 if X = ±1

∴

`P(Y =0) = P(X = 0) =(1)/(3)`

`P(Y = 1) = P(X = -1 \ or \ 1) \\ \\ = P(X = -1) + P(X = 1)`

`= (1)/(2)+ (1)/(6)`

`=(3+1)/(6)`

`= (2)/(3)`

b)

`E(|X|) = \sum |x| P(X=x) = ( 1 * (1)/(2)) + ( 0* (1)/(3)) + ( 1 * (1)/(6))`

`= (2)/(3)`