Question

$X$ is a discrete random variable taking values $0,1,2$ with $P(X=0)=0.25$, $P(X=1)=0.5$, and $P(X=2)=0.25$. With $E[ . ]$ denoting expectation, compute $E[X]-E[X^2]$ (rounded off to one decimal place).

Hint:

For discrete $X$: $E[g(X)]=\sum_x g(x)\,P(X=x)$. Compute $E[X]$ and $E[X^2]$ separately, then subtract.

Answer 1

Correct Answer: -0.5

\[ E[X]=0(0.25)+1(0.5)+2(0.25)=1.0, E[X^2]=0^2(0.25)+1^2(0.5)+2^2(0.25)=1.5. \] Therefore, \[ E[X]-E[X^2]=1.0-1.5=\boxed{-0.5}. \]