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

  • Notes
  • 311
  • 100% (1) 1 out of 1 people found this document helpful

This preview shows page 173 - 185 out of 311 pages.

n 1 ! ... n m + 1 ! such orderings.
Image of page 173

Subscribe to view the full document.

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 ) .
Image of page 174
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?
Image of page 175

Subscribe to view the full document.

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.
Image of page 176
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 =
Image of page 177

Subscribe to view the full document.

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
Image of page 178
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 .
Image of page 179

Subscribe to view the full document.

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 ) =
Image of page 180
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
Image of page 181

Subscribe to view the full document.

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
Image of page 182
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 .
Image of page 183

Subscribe to view the full document.

Example 1.30 Let N A , N B and N F
Image of page 184
Image of page 185

{[ snackBarMessage ]}

Get FREE access by uploading your study materials

Upload your study materials now and get free access to over 25 million documents.

Upload now for FREE access Or pay now for instant access
Christopher Reinemann
"Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

Ask a question for free

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern