Excercises

Covariance rules

Simplify:

$$\mathrm{Cov}[2X+5Y+1,3Y+8W+X+9]$$

Dependent but uncorrelated variables

Let $X$ be given with possible values $\{-\mathrm{1,0},+1\}$ and PMF given (uniformly) by $P_X(k)=1/3$ for all three possible $k$. Let $Y=X^2$.

Show that $X$ and $Y$ are dependent but uncorrelated.

Hint: To speed the calculation, notice that $X^3=X$.