3.5.1 Multinomial Flashcards
(14 cards)
What is the multinomial distribution?
A generalization of the binomial distribution used when each trial has more than two possible outcomes.
When should you use a multinomial distribution?
When you perform n independent trials, each with k possible outcomes, and you want the joint distribution of the counts of each outcome.
What is the joint PMF formula for the multinomial distribution?
P(X₁ = x₁, …, X_k = x_k) = [n! / (x₁! x₂! … x_k!)] * (p₁)^{x₁} * (p₂)^{x₂} * … * (p_k)^{x_k}, where Σxᵢ = n and Σpᵢ = 1.
What do x₁, …, x_k represent in the multinomial formula?
The number of times each outcome occurs in n trials.
What does the multinomial coefficient count?
The number of ways to partition n items into k categories of sizes x₁, x₂, …, x_k.
What are the conditions for the multinomial formula?
- n is fixed
- Outcomes are mutually exclusive
- Trials are independent
- Probabilities pᵢ are constant and sum to 1.
What is the marginal distribution of a single outcome Xᵢ in a multinomial?
Binomial(n, pᵢ).
What is the mean and variance of Xᵢ in a multinomial?
E[Xᵢ] = n * pᵢ, Var(Xᵢ) = n * pᵢ * (1 - pᵢ).
What is the covariance between Xᵢ and Xⱼ in a multinomial?
Cov(Xᵢ, Xⱼ) = -n * pᵢ * pⱼ (for i ≠ j).
Why is the covariance between categories negative in a multinomial?
Because an increase in one outcome implies fewer trials are left for others — their counts are not independent.
Why can’t you always use the binomial shortcut in multinomial problems?
If more than two outcomes have non-zero counts, the binomial shortcut doesn’t apply — use full multinomial formula.
In Example 3.5.1, why was the binomial used for red marbles?
Because only the count of red marbles mattered — the other categories were grouped as ‘not red.’
What’s a good way to structure a multinomial problem?
Step 1: Identify n and k
Step 2: Define counts x₁, …, x_k
Step 3: Identify probabilities p₁, …, p_k
Step 4: Plug into the multinomial PMF.
What is a multinomial experiment?
A set of n independent trials, each resulting in exactly one of k mutually exclusive outcomes.
