2001s_2

# N 1 n m 1 such orderings p n 1 n 1 n m n m n n 1 n m

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n 1 ! ... n m + 1 ! such orderings.

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P ( N 1 = n 1 , . . . , N m = n m ) = ( n ! n 1 ! ... n m + 1 ! p n 1 1 . . . p n m + 1 m + 1 , n 1 + . . . + n m n 0 otherwise. If m = 1 the multinomial distribution is just the binomial distribution, i.e. N 1 Bin ( n , p 1 ) , which has mean np 1 and variance np 1 ( 1 - p 1 ) .
P ( N 1 = n 1 , . . . , N m = n m ) = ( n ! n 1 ! ... n m + 1 ! p n 1 1 . . . p n m + 1 m + 1 , n 1 + . . . + n m n 0 otherwise. Example 1.29 Suppose that a bag contains five red, five black and five yellow balls and that three balls are drawn at random with replacement. What is the probability that there is one of each colour?

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Example 1.29 Suppose that a bag contains five red, five black and five yellow balls and that three balls are drawn at random with replacement. What is the probability that there is one of each colour? I Sampling with replacement.
Example 1.29 Suppose that a bag contains five red, five black and five yellow balls and that three balls are drawn at random with replacement. What is the probability that there is one of each colour? I Sampling with replacement. I p red =

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Example 1.29 Suppose that a bag contains five red, five black and five yellow balls and that three balls are drawn at random with replacement. What is the probability that there is one of each colour? I Sampling with replacement. I p red = 5 15 = 1 / 3
Example 1.29 Suppose that a bag contains five red, five black and five yellow balls and that three balls are drawn at random with replacement. What is the probability that there is one of each colour? I Sampling with replacement. I p red = 5 15 = 1 / 3 I p black = p yellow = 1 3 .

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Example 1.29 Suppose that a bag contains five red, five black and five yellow balls and that three balls are drawn at random with replacement. What is the probability that there is one of each colour? I Sampling with replacement. I p red = 5 15 = 1 / 3 I p black = p yellow = 1 3 . P ( N red = N black = N yellow = 1 ) =
Example 1.29 Suppose that a bag contains five red, five black and five yellow balls and that three balls are drawn at random with replacement. What is the probability that there is one of each colour? I Sampling with replacement. I p r = 5 15 = 1 / 3 I p b = p y = 1 3 . P ( N r = n r , N b = n b , N y = n y ) = n ! n r ! n b ! n y ! p n r r p n b b p n y y

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Example 1.29 Suppose that a bag contains five red, five black and five yellow balls and that three balls are drawn at random with replacement. What is the probability that there is one of each colour? I Sampling with replacement. I p r = 5 15 = 1 / 3 I p b = p y = 1 3 . P ( N r = n r , N b = n b , N y = n y ) = n ! n r ! n b ! n y ! p n r r p n b b p n y y = 3 ! 1 ! 1 ! 1 ! 1 3 1 1 3 1 1 3 1 = 6 1 27 = 2 9
Marginal distribution of N i . Clearly N i can be regarded as the number of successes in n independent Bernoulli trials if we define success to be individual is of type i . Thus N i has a binomial distribution, N i Bin ( n , p i ) , with mean np i and variance np i ( 1 - p i ) . Example 1.30 Let N A , N B and N F be the numbers of A grades, B grades and fails respectively amongst a class of 100 students. Suppose that generally 5% of students achieve grade A, 30% grade B and that 5% fail. Write down the joint distribution of N A , N B and N F and find the marginal distribution of N A .

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Example 1.30 Let N A , N B and N F

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