Conditional Expectation of two Multiplied Variables

Question

Given that R and S have joint probability mass function:

$\qquad\begin{array}{cc}& S\\ R &\begin{array} {r|rrrr|} & -1& 0 & 1 & 2 \\\hline 0 & \tfrac 1{10} & \tfrac 1{10} & 0 & 0 \\ 1 & \tfrac 1{20} & \tfrac 1{10} & \tfrac 1{10} & 0 \\ 2 & 0 & \tfrac 1{20} & \tfrac 1{10} & \tfrac 1{20} \\ \bbox[yellow, 2pt]3 & \bbox[yellow, 2pt]0 & \bbox[yellow, 2pt]0 & \bbox[yellow, 1pt]{\tfrac 1{10}} & \bbox[yellow, 1pt]{\tfrac 1{10}} \\ 4 & 0 & 0 & 0 & \tfrac 1{5} \end{array}\end{array}$

Calculate the expectation of RS given that R = 3.

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Were trying to find E(RS|R=3). I know the first step should be finding P(R=3) and then letting each probability in R=3 be divided by the marginal probability of R=3. But I'm not sure what each of these should be multiplied with since the conditional probability were trying to find is RS instead of just one random variable.

Answer 1

I know the first step should be finding P(R=3) and then letting each probability in R=3 be divided by the marginal probability of R=3. But I'm not sure what each of these should be multiplied with since the conditional probability were trying to find is RS instead of just one random variable.

You want the expectation of the product of the values, $RS$, given the condition $\{R=3\}$, so the conditional probabilities should be multiplied by the product of the values whenever the condition happens.

That is simply $\sum_s 3s\mathsf P(S=s\mid R=3)$

$$\begin{align}\mathsf E(RS\mid R=3) ~=~& \dfrac{\mathsf E(3S~\mathbf 1_{R=3})}{\mathsf P(R=3)}\\[1ex] ~=~& \dfrac{3\sum_{s} s\,\mathsf P(R=3, S=s)}{\sum_{s} \mathsf P(R=3, S=s)} & =~& 3\sum_{s}s\,\mathsf P(S=s\mid R=3)\\[1ex]~=~& \dfrac{3\big({-1\cdot 0+0\cdot 0+}1\cdot\tfrac 1{10}+2\cdot\tfrac 1{10}\big)}{{0+0+}\tfrac 1{10}+\tfrac 1{10}} \\~=~& \dfrac{9}{2} \end{align}$$