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Competitive Stats
Q43Section I (Objective (Q1–Q80))43 of 80

Distribution Theory — conditional mean in multinomial

Consider the following for the next four (04) items: Let (X1,X2,X3)(X_1, X_2, X_3) be a random vector with the joint pmf given by:

P(X1=x1,X2=x2,X3=x3)={50!x1!x2!x3!(50−x1−x2−x3)!(14)x1(14)x2(14)x3(14)50−x1−x2−x3if x1+x2+x3≤50 0otherwise\mathbb{P}(X_1 = x_1, X_2 = x_2, X_3 = x_3) = \begin{cases} \frac{50!}{x_1!x_2!x_3!(50-x_1-x_2-x_3)!}\left(\frac{1}{4}\right)^{x_1}\left(\frac{1}{4}\right)^{x_2}\left(\frac{1}{4}\right)^{x_3}\left(\frac{1}{4}\right)^{50-x_1-x_2-x_3} & \text{if } x_1+x_2+x_3 \le 50 \ 0 & \text{otherwise} \end{cases}

What is E(X1∣X2=5)\mathbb{E}(X_1 \mid X_2 = 5) equal to?

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