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Chapter 3: Problem 10

Let \(X_{1}, X_{2}, \ldots, X_{k-1}\) have a multinomial distribution. (a) Find the mgf of \(X_{2}, X_{3}, \ldots, X_{k-1}\). (b) What is the pmf of \(X_{2}, X_{3}, \ldots, X_{k-1} ?\) (c) Determine the conditional pmf of \(X_{1}\) given that \(X_{2}=x_{2}, \ldots, X_{k-1}=x_{k-1}\). (d) What is the conditional expectation \(E\left(X_{1} \mid x_{2}, \ldots, x_{k-1}\right) ?\)

Short Answer

Expert verified
The Moment Generating Function (mgf) of variables \(X_{2}, X_{3}, ..., X_{k-1}\) is \((p_{2} + ... + p_{k})^{n}\). The Probability Mass Function (pmf) is \(\frac{n!}{x_{2}!...x_{k}!} p_{2}^{x_{2}}...p_{k}^{x_{k}}\). The conditional pmf of \(X_{1}\) given \(X_{2}=x_{2}, ..., X_{k-1}=x_{k-1}\) is \(\frac{n!}{(n - x_{2} - ... - x_{k-1})!x_{2}!...x_{k-1}!} p_{1}^{n - x_{2} - ... - x_{k-1}}p_{2}^{x_{2}}...p_{k}^{x_{k}}\). The conditional expectation \(E(X_{1}| x_{2},...,x_{k})\) is \(n - x_{2} - ... - x_{k-1}\).

Step by step solution

01

Calculate the Moment Generating Function (mgf)

The mgf of a random variable \(X\) with a multinomial distribution is \(M(t) = (p_{1}e^{t} + p_{2} + ... + p_{k})^{n}\). Considering the given variables \(X_{2}, X_{3}, ..., X_{k-1}\), the mgf would be the same but without the factor \((p_{1}e^{t})\), which corresponds to \(X_{1}\). Thus the mgf would be \((p_{2} + ... + p_{k})^{n}\).
02

Calculate the Probability Mass Function (pmf)

The pmf of a random variable \(X\) with a multinomial distribution is given by \(P(X_{1}=x_{1}, X_{2}=x_{2}, ..., X_{k}=x_{k}) = \frac{n!}{x_{1}!x_{2}!...x_{k}!} p_{1}^{x_{1}}p_{2}^{x_{2}}...p_{k}^{x_{k}}\). For the given variables \(X_{2}, X_{3}, ..., X_{k-1}\), the pmf would be the same but without the factors related to \(X_{1}\). Therefore, the pmf is \(\frac{n!}{x_{2}!...x_{k}!} p_{2}^{x_{2}}...p_{k}^{x_{k}}\)
03

Compute the conditional pmf of \(X_{1}\)

Given \(X_{2}=x_{2}, ..., X_{k-1}=x_{k-1}\), the conditional pmf of \(X_{1}\) is derived from the multinomial pmf, with \(X_{1} = n - x_{2} - ... - x_{k-1}\) since the total observations must sum to \(n\). The conditional pmf is then \(\frac{n!}{(n - x_{2} - ... - x_{k-1})!x_{2}!...x_{k-1}!} p_{1}^{n - x_{2} - ... - x_{k-1}}p_{2}^{x_{2}}...p_{k}^{x_{k}}\)
04

Calculate the Conditional Expectation

The expectation \(E(X_{1}| x_{2},...,x_{k})\) is the weighted average, or expected value, of \(X_{1}\) given all other variables. It's calculated as \(E(X_{1}| x_{2}, ..., x_{k}) = n - x_{2} - ... - x_{k-1}\), which is based on the constraint that the total observations \(n = x_{1} + x_{2} + ... + x_{k}\)

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