Question

If \((X,Y)\) is a random vector with the joint probability mass function

Hint:

When finding conditional variance, first identify the conditional distribution. Adding a constant does not change variance.

Answer 1

Correct Answer: 0.75

Step 1: Fix $X=2$ and shift $Y$.
For $X=2$, the conditional mass is proportional to $3^{3-y}$ for $y=2,3,\dots$. Let $k=y-2$, so it is proportional to $3^{1-k}$, $k\ge0$.

Step 2: Normalise.
$\sum_{k\ge0}3^{1-k}=3\cdot\frac1{1-1/3}=\frac92$, so $P(K=k)=\frac23\left(\frac13\right)^k$.

Step 3: Recognise the distribution.
This is geometric on $0,1,2,\dots$ with $p=\frac23$, $q=\frac13$.

Step 4: Variance.
$\mathrm{Var}(K)=\frac{q}{p^2}=\frac{1/3}{4/9}=\frac34=0.75$, and $Y=K+2$ has the same variance.

Step 5: Conclude.
\[ \boxed{0.75} \]