I will rate asap. Thanks Problem 3. Let (Xi, X2, X3, X4) be Multinomial(n, 4,1/6, 1/3,1/8,3/8)....

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Problem 3. Let (Xi, X2, X3, X4) be Multinomial(n, 4,1/6, 1/3,1/8,3/8). Derive the joint mass function of the pair (X3, X4). You should be able to do this with almost no computation.

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Answer 1

Answer #1

We are given here that:
P(X₃ = 1) = 1/8 and P(X₄ = 1) = 3/8

Therefore, P(X₃ = X₄ = 0) = 1 - (1/8) - (3/8) = 1/2

Therefore, the joint mass function of X₃, X₄ here is obtained as:

P(X₃ = x₃, X₄ = x₄) = Probability of getting x₃ instances of X₃, x₄ instances of X₄ and (4 - x₃ - x₄) instances of anything other than x₃ / x₄.

Therefore, the probability now is obtained here as:

$P(X_3= x_3, X_4 = x_4) = \binom{4}{x_3}(1/8)^{x_3}\binom{4 - x_3}{x_4}(3/8)^{x_4}*(1/2)^{4 - x_3 - x_4}$

This is the required joint PDF for X₃ , X₄

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Answer 2

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