Multinomial Distribution

The multinomial distribution is an extension of the binomial distribution involving joint probabilities. It involves a similar statistical experiment, but this time there are more than two possible outcomes. Specifically, each trial can result in any of the k events E₁ , E₂ , ...., E_k , with respective probabilities p₁ , p₂ , .... , p_k. In this case, the multinomial distribution is the joint probability distribution of the set of random variables X₁ , X₂ , ...., X_k , where X_i is the number of occurrences of E_i , i = 1, 2, ...., k, in n independent trials. It has a probability mass function of the following form:

The multinomial term epresents the number of ways distribute x₁ outcomes of E₁ , x₂ outcomes of E₂ , . . . x_k outcomes of E_kn trials. among

The term is the probability that there are x₁ outcomes of E₁, x₂ outcomes of E₂ , x_k outcomes of E_k.

The products of these two terms is the probability that in n trials, there are x₁ outcomes for E₁, x₂ outcomes for E₂,. . . x_k outcomes for E_k.

EX. On average, Mark has a 50 % probability of not getting a hit during an at-bat opportunity. His probabilities are 12.5 % for a single, 10 % for a double, 2.5 % for a triple and 5 % for a home run. He gets a walk 20% of the time. The probability distribution for the number of each type of hit, as well as outs and walks, in n at-bats is modeled as follows:

Let random variable

The probability that Mark hits for the cycle (gets a single, double, triple and home run) in the next four at-bats is

p(0, 1, 1, 1, 1, 0; 4)