Question

1) Let the five r.v.'s $X_1, X_2, X_3, X_4, X_5$ have the multinomial distribution with parameters 15 and \frac{1}{15}, \frac{3}{15}, \frac{2}{15}, \frac{5}{15}, \frac{4}{15}. That is, $(X_1, X_2, X_3, X_4, X_5) \sim M(15; \frac{1}{15}, \frac{3}{15}, \frac{2}{15}, \frac{5}{15}, \frac{4}{15}). (a) What is the joint distribution of $X_1, X_3, X_5$? (4%) (b) What is the conditional joint distribution of $X_1, X_3, X_5$, given $X_2 = 3, X_4 = 5$? (4%) (c) Find the joint m.g.f. of $X_2, X_4$. (Hint: The joint m.g.f. of $X_1, X_2, X_3, X_4, X_5 = ?)$ (4%) (d) Find $Cov(X_2, X_4)$ and $\rho(X_2, X_4)$. (6%)

          1) Let the five r.v.'s $X_1, X_2, X_3, X_4, X_5$ have the multinomial distribution with parameters 15 and \frac{1}{15}, \frac{3}{15}, \frac{2}{15}, \frac{5}{15}, \frac{4}{15}. That is, $(X_1, X_2, X_3, X_4, X_5) \sim M(15; \frac{1}{15}, \frac{3}{15}, \frac{2}{15}, \frac{5}{15}, \frac{4}{15}).
(a) What is the joint distribution of $X_1, X_3, X_5$?
(4%)
(b) What is the conditional joint distribution of $X_1, X_3, X_5$, given $X_2 = 3, X_4 = 5$?
(4%)
(c) Find the joint m.g.f. of $X_2, X_4$. (Hint: The joint m.g.f. of $X_1, X_2, X_3, X_4, X_5 = ?)$ 
(4%)
(d) Find $Cov(X_2, X_4)$ and $\rho(X_2, X_4)$.
(6%)

1) Let the five r.v.'s X1, X2, X3, X4, X5 have the multinomial distribution with parameters 15 and (1)/(15), (3)/(15), (2)/(15), (5)/(15), (4)/(15). That is, (X1, X2, X3, X4, X5) ∼ M(15; (1)/(15), (3)/(15), (2)/(15), (5)/(15), (4)/(15)).
(a) What is the joint distribution ofX1, X3, X5?
(4%)
(b) What is the conditional joint distribution ofX1, X3, X5, givenX2 = 3, X4 = 5?
(4%)
(c) Find the joint m.g.f. ofX2, X4. (Hint: The joint m.g.f. ofX1, X2, X3, X4, X5 = ?)(4%)
(d) FindCov(X2, X4)andρ(X2, X4).
(6%)

Added by David V.

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