Cormen Edition 3 Exercise 8.4 Question 3 (Page No. 204)

Question

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 $E^2[X]$?

Answer 1

Sampling out the scenario we get

${HH,HT,TH,TT}$ each with probability $\frac{1}{4}$

 

$P(X=no. of heads)$

$E[X]\Rightarrow 2*\frac{1}{4} + 1*\frac{1}{4}+1*\frac{1}{4}+0*\frac{1}{4}$

$\Rightarrow \frac{2}{4}+\frac{2}{4}\Rightarrow 1$

 $E[X]=1$;  $E^{2}[X]\Rightarrow 1$

 

Now calculating for $E[X^{2}]$

$E[X^{2}]\Rightarrow 2^{2}*\frac{1}{4}+1^{2}*\frac{1}{4}+1^{2}*\frac{1}{4}+0^{2}*\frac{1}{4}$

$\Rightarrow 1+\frac{1}{2}\Rightarrow 1.5$

so $E[X^{2}]$$=1.5$