Suppose we consider a trial that can have l possible outcomes (say, rolling an l-sided die)....

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Suppose we consider a trial that can have l possible outcomes (say, rolling an l-sided die). Each of the outcomes occurs with some probability P₁, P2, ..., pe with p₁+p2+...+pe = 1. Suppose we have n independent trials. Let Z (Z₁,... ..., Ze) be the random vector specifying the number of times each outcome occurs. (a) Show that we have n ki P₁ P(Z = (k₁,..., kt)) = (k₁, ..., k.). p^² ke 2 ke Pe = n! kı k₁!... k₁¹... pke, where k₁+...+ke = n. This is known as the the Multinomial distribution. (b) Show that the marginal distribution of each Z; is given by a binomial distribution with parameter pį. (Hint: you can do this without computing any nasty sums.) Suppose we consider a trial that can have l possible outcomes (say, rolling an l-sided die). Each of the outcomes occurs with some probability P₁, P2, ..., pe with p₁+p2+...+pe = 1. Suppose we have n independent trials. Let Z (Z₁,... ..., Ze) be the random vector specifying the number of times each outcome occurs. (a) Show that we have n ki P₁ P(Z = (k₁,..., kt)) = (k₁, ..., k.). p^² ke 2 ke Pe = n! kı k₁!... k₁¹... pke, where k₁+...+ke = n. This is known as the the Multinomial distribution. (b) Show that the marginal distribution of each Z; is given by a binomial distribution with parameter pį. (Hint: you can do this without computing any nasty sums.)

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a The multinomial distribution describes the probability of observing a specific combination of outc... View the full answer