Calculating Expected Value and Variance of Coin Toss Results

[pawlo392]

A coin had tossed three times. Let $X$- number of tails and $Y$- number of heads. Find the expected value and variance $Z=XY$.
My solution:
We know, that $Y=3-X$, so $Z=(3-X)X$ for $X=0,1,2,3$.
$Z=2$ for $X=1,2$ and $Z=0$ for $X=3,0$
So, $E(Z)=E((3-X)X))= 2 \cdot ⅜ +2 \cdot ⅜.$
Because we're interested in events :
1 head and 2 tails, and 1 tail and 2 heads.
And $E(Z^2)=4 \cdot⅜ +4 \cdot ⅜$. What do You think?

 

[FactChecker]

That looks good to me.