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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
, ...., Ek , with respective probabilities p1
, p2 , .... , pk. In this case, the
multinomial distribution is the joint probability distribution of
the set of random variables X1 , X2 , ....,
Xk , where Xi is the number of occurrences
of Ei , i = 1, 2, ...., k, in n
independent trials. It has a probability mass function of the
following form:
The multinomial term  represents the number
of ways distribute x1 outcomes of E1 , x2
outcomes of E2 , . . . xk outcomes of Ekn trials.
among
The term  is the probability
that there are x1 outcomes of E1, x2
outcomes of E2 , xk outcomes of Ek.
The products of these two terms is the probability that in n
trials, there are x1 outcomes for E1, x2
outcomes for E2,. . . xk outcomes for Ek.
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
| X1 |
= number of outs |
p1 = 0.500 |
| X2 |
= number of singles |
p2 = 0.125 |
| X3 |
= number of doubles |
p3 = 0.100 |
| X4 |
= number of triples |
p4 = 0.025 |
| X5 |
= number of home runs |
p5 = 0.050 |
| X6 |
= number of walks |
p6 = 0.200 |
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)
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