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Multinomial Distribution

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:

6dpd011

The multinomial term 6dpd012 epresents the number of ways distribute x1 outcomes of E1 , x2 outcomes of E2 , . . . xk outcomes of Ekn trials. among

The term 6dpd013 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

6dpd014

6dpd015

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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Statistics [1]
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