Question

The joint probability mass function of \(X\) and \(Y, p(x, y)\), is given by $$ \begin{array}{ll} p(1,1)=\frac{1}{9}, & p(2,1)=\frac{1}{3}, & p(3,1)=\frac{1}{9} \\ p(1,2)=\frac{1}{9}, & p(2,2)=0, & p(3,2)=\frac{1}{18} \\ p(1,3)=0, & p(2,3)=\frac{1}{6}, & p(3,3)=\frac{1}{9} \end{array} $$ Compute \(E[X \mid Y=i]\) for \(i=1,2,3\).

Short answer

The conditional expectations \(E[X \mid Y=i]\) for \(i=1,2,3\) are as follows: For \(Y=1\): \(E[X \mid Y=1] =\frac{13}{6}\) For \(Y=2\): \(E[X \mid Y=2] =\frac{7}{4}\) For \(Y=3\): \(E[X \mid Y=3] =\frac{20}{9}\)

Step-by-step solution

Calculate the marginal probability of Y for each value i

Since we need to compute the conditional probabilities for each value of i, we first need to calculate the marginal probability of Y=i. Then marginal probability of Y is given by summing the joint probabilities of each combination that has Y=i. For Y=1, the marginal probability is calculated by summing the probabilities for (1, 1), (2, 1), and (3, 1) as follows: \(P(Y=1) = p(1,1) + p(2,1) + p(3,1) = \frac{1}{9} + \frac{1}{3} + \frac{1}{9}\) For Y=2, the marginal probability is calculated by summing the probabilities for (1, 2), (2, 2), and (3, 2) as follows: \(P(Y=2) = p(1,2) + p(2,2) + p(3,2) = \frac{1}{9} + 0 + \frac{1}{18}\) For Y=3, the marginal probability is calculated by summing the probabilities for (1, 3), (2, 3), and (3, 3) as follows: \(P(Y=3) = p(1,3) + p(2,3) + p(3,3) = 0 + \frac{1}{6} + \frac{1}{9}\)

Calculate the conditional probability P(X=x | Y=i) for each value of i

Now, we will calculate the conditional probability for each combination using the formula: \(P(X=x \mid Y=i) = \frac{p(x,i)}{P(Y=i)}\) For Y=1: \(P(X=1 \mid Y=1) = \frac{p(1,1)}{P(Y=1)}\) \(P(X=2 \mid Y=1) = \frac{p(2,1)}{P(Y=1)}\) \(P(X=3 \mid Y=1) = \frac{p(3,1)}{P(Y=1)}\) For Y=2: \(P(X=1 \mid Y=2) = \frac{p(1,2)}{P(Y=2)}\) \(P(X=2 \mid Y=2) = \frac{p(2,2)}{P(Y=2)}\) \(P(X=3 \mid Y=2) = \frac{p(3,2)}{P(Y=2)}\) For Y=3: \(P(X=1 \mid Y=3) = \frac{p(1,3)}{P(Y=3)}\) \(P(X=2 \mid Y=3) = \frac{p(2,3)}{P(Y=3)}\) \(P(X=3 \mid Y=3) = \frac{p(3,3)}{P(Y=3)}\)

Compute the conditional expectation E[X | Y=i]

Now we will compute the conditional expectation for each Y=i using the following formula: \(E[X \mid Y=i] = \sum_{x} x \cdot P(X=x \mid Y=i)\) For Y=1: \(E[X \mid Y=1] = 1 \cdot P(X=1 \mid Y=1) + 2 \cdot P(X=2 \mid Y=1) + 3 \cdot P(X=3 \mid Y=1)\) For Y=2: \(E[X \mid Y=2] = 1 \cdot P(X=1 \mid Y=2) + 2 \cdot P(X=2 \mid Y=2) + 3 \cdot P(X=3 \mid Y=2)\) For Y=3: \(E[X \mid Y=3] = 1 \cdot P(X=1 \mid Y=3) + 2 \cdot P(X=2 \mid Y=3) + 3 \cdot P(X=3 \mid Y=3)\) Using the conditional probabilities calculated in Step 2, compute the conditional expectations for each value of Y (Y=1,2,3).