8.4-3

8.4-3 Let X be a random variable that is equal to the number of heads in two flips of a fair coin. What is $E[X^2]$ ? What is $E^2[X] $?

Lets calculate $X^2$
we get HT,TH and HH possibilities where get at least one head
The probability of one head is 1/2
The probability of two heads is 1/4 since X is no of heads
$X^2 = 1^2 + 2^2 $
$E[X^2] = 1 . 1/2 + 4 . 1/4$
$E[X^2] = 1.5 $
What is $E^2[X] $?
$$ E[X] = 1. 1/2+2.1/4 $$ $$ E[X] = 1$$ $$ E[E[X] ] = E[1]$$ $$ E^2[X] ] = 1$$