Question

Let $X_1,X_2,\dots ,X_{k-1}$ have a multinomial distribution. (a) Find the mgf of $X_2,X_3,\dots ,X_{k-1}$. (b) What is the pmf of $X_2,X_3,\dots ,X_{k-1}?$ (c) Determine the conditional pmf of $X_1$ given that $X_2=x_2,\dots ,X_{k-1}=x_{k-1}$. (d) What is the conditional expectation $E(X_1\mid x_2,\dots ,x_{k-1})$ ?

Short answer

1. MGF: $M(t)={(1-p_1+\sum _{i=2}^{k-1}{p}_i{e}^{t_i})}^n$ 2. PMF: $P(X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})=\frac{n!}{x_2!x_3!...x_{k-1}!}{p}_2^{x_2}{p}_3^{x_3}...p_{k-1}^{x_{k-1}}$ 3. Conditional PMF: $P(X_1=x_1|X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})$. 4. Conditional Expectation: $E[X_1|X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1}]$.

Step-by-step solution

Find the MGF

The moment generating function (mgf) of $X_2,X_3,...,X_{k-1}$ can be represented as $$M(t)={(1-p_1+\sum _{i=2}^{k-1}{p}_i{e}^{t_i})}^n$$where $p_i$ are the probabilities associated with these variables, $t_i$ are variables for the generating function and n is the total number of trials.

Determine the PMF

The probability mass function (pmf) of $X_2,X_3,...,X_{k-1}$ can be expressed as $$P(X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})=\frac{n!}{x_2!x_3!...x_{k-1}!}{p}_2^{x_2}{p}_3^{x_3}...p_{k-1}^{x_{k-1}}$$where $x_2,x_3,..,x_{k-1}$ outlets their respective outcomes.

Determine the Conditional PMF

The conditional pmf of $X_1$ given $X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1}$ is $$P(X_1=x_1|X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})=\frac{P(X_1=x_1,X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})}{P(X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})}$$where $x_1$ is the value that $X_1$ gets.

Find the Conditional Expectation

The expectation of $X_1$ given $X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1}$ is $$E[X_1|X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1}]=\sum _{x_1=0}^{n-\sum _{i=2}^{k-1}{x}_i}{x}_{1}P(X_1=x_1|X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})$$This represents the expected value of $X_1$ given the specified conditions.

Key concepts

Moment Generating Function (MGF)

The Moment Generating Function, commonly abbreviated as MGF, is a crucial tool in probability and statistics. It serves as a way to characterize the distribution of a random variable. For a vector of random variables such as $X_2,X_3,...,X_{k-1}$, the MGF is expressed as $M(t)={(1-p_1+\sum _{i=2}^{k-1}{p}_i{e}^{t_i})}^n$.

This formula gives us a compact representation of the entire distribution. In the MGF, $p_i$ are the probabilities linked to each random variable. Meanwhile, $t_i$ are parameters for the generating function, acting as variables which we integrate over.

The exponent $n$ represents the total number of trials, often linked to the size of our dataset in probabilistic models. By taking derivatives of the MGF, we can extract moments like the mean and variance. This makes the MGF invaluable, since calculating higher moments directly can be tedious. As such, MGFs are heavily relied upon when dealing with complex distributions like the multinomial distribution.

Probability Mass Function (PMF)

The Probability Mass Function, or PMF, is a fundamental concept in probability theory that helps us determine the likelihood of discrete random variables. For a collection of variables like $X_2,X_3,...,X_{k-1}$, the PMF is calculated using the formula:$$P(X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})=\frac{n!}{x_2!x_3!...x_{k-1}!}{p}_2^{x_2}{p}_3^{x_3}...p_{k-1}^{x_{k-1}}$$
This expression predicts the probability of observing a specific combination of outcomes. Here, the factorials $n!$ and $x_i!$ are part of combinatorial mathematics, showing how outcomes can be arranged and considering their order.

The terms $p_i^{x_i}$ indicate the probability of each specific outcome raised to the power of the number of times it occurs. This exponential form captures the way probabilities compound when multiple independent variables are involved. Understanding the PMF is crucial since it allows us to grasp how likely different outcomes are, providing a detailed picture of what we might expect to observe in real life.

Conditional Expectation

Conditional Expectation is an important concept in statistics and is used to determine the expected value of a random variable given that certain conditions are met. In our case, we want to find the expectation of $X_1$ given that $X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1}$.

This is expressed mathematically as:$$E[X_1|X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1}]=\sum _{x_1=0}^{n-\sum _{i=2}^{k-1}{x}_i}{x}_{1}P(X_1=x_1|X_2=x_2,X_3=x_3,...,X_{k-1}=x_{k-1})$$This equation sums up all possible values of $X_1$, weighted by their conditional probabilities.

In simpler terms, conditional expectation provides insights into what we might anticipate, given specific known information. It's like having a refined prediction based on certain events occurring. This becomes especially meaningful in practical situations where past events or data guide future predictions, making it an essential tool for statisticians and data analysts.