Chapter 5: Special Random Variables 17. IfX is a Poisson random variable with mean 2, show that P{X=i} first increases and then decreases as i increases, reaching its maximum value when i is the largest 000 18. A contractor purchases a shipment of 100 transistors. It is his policy to test 10 of these transistors and to keep the shipment only if at least 9 of the working condition. If the shipment contains 20 defective transistors, what is the integer less than or equal to 2. probability it will be kept? 19. Let X denote a hypergeometric random variable with parameters n, m, and k. That is, ()C) (w) i = 0, 1,..., min(k, n) P{X = i} = %3D (a) Derive a formula for P{X = i} in terms of P{X =i – 1}. (b) Use part (a) to compute P{X = i} for i = 0, 1, 2, 3, 4, 5 when n = m = 10. k = 5, by starting with P{X = 0}. (c) Based on the recursion in part (a), write a program to compute the hyper- %3D %3D 10, %3D %3D %3D geometric distribution function. (d) Use your program from part (c) to compute P{X < 10} when n = m= 30, k = 15. 20. Independent trials, each of which is a success with probability p, are successively performed. Let X denote the first trial resulting in a success. That is, X will equal k if the first k-1 trials are all failures and the kth a success. X is called a geometric random variable. Compute (a) P{X= k}, k = 1, 2, ...; (b) E[X]. %3D Let Y denote the number of trials needed to obtain r successes. Y is called a negative binomial random variable. Compute (c) P(Y = k}, k= r,r+1,.... (Hint: In order for Y to equal k, how many successes must result in the first k: trials and what must be the outcome of trial k?) (d) Show that E[Y] = rlp (Hint: Write Y = Y1++ Y, where Y; is the number of trials needed to go from a total of i-1 to a total of i successes.)
Chapter 5: Special Random Variables 17. IfX is a Poisson random variable with mean 2, show that P{X=i} first increases and then decreases as i increases, reaching its maximum value when i is the largest 000 18. A contractor purchases a shipment of 100 transistors. It is his policy to test 10 of these transistors and to keep the shipment only if at least 9 of the working condition. If the shipment contains 20 defective transistors, what is the integer less than or equal to 2. probability it will be kept? 19. Let X denote a hypergeometric random variable with parameters n, m, and k. That is, ()C) (w) i = 0, 1,..., min(k, n) P{X = i} = %3D (a) Derive a formula for P{X = i} in terms of P{X =i – 1}. (b) Use part (a) to compute P{X = i} for i = 0, 1, 2, 3, 4, 5 when n = m = 10. k = 5, by starting with P{X = 0}. (c) Based on the recursion in part (a), write a program to compute the hyper- %3D %3D 10, %3D %3D %3D geometric distribution function. (d) Use your program from part (c) to compute P{X < 10} when n = m= 30, k = 15. 20. Independent trials, each of which is a success with probability p, are successively performed. Let X denote the first trial resulting in a success. That is, X will equal k if the first k-1 trials are all failures and the kth a success. X is called a geometric random variable. Compute (a) P{X= k}, k = 1, 2, ...; (b) E[X]. %3D Let Y denote the number of trials needed to obtain r successes. Y is called a negative binomial random variable. Compute (c) P(Y = k}, k= r,r+1,.... (Hint: In order for Y to equal k, how many successes must result in the first k: trials and what must be the outcome of trial k?) (d) Show that E[Y] = rlp (Hint: Write Y = Y1++ Y, where Y; is the number of trials needed to go from a total of i-1 to a total of i successes.)
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Q 20
![Chapter 5: Special Random Variables
17. IfX is a Poisson random variable with mean 2, show that P{X=i} first increases
and then decreases as i increases, reaching its maximum value when i is the largest
000
18. A contractor purchases a shipment of 100 transistors. It is his policy to test 10
of these transistors and to keep the shipment only if at least 9 of the
working condition. If the shipment contains 20 defective transistors, what is the
integer less than or equal to 2.
probability it will be kept?
19. Let X denote a hypergeometric random variable with parameters n, m, and k.
That is,
()C)
(w)
i = 0, 1,..., min(k, n)
P{X = i} =
%3D
(a) Derive a formula for P{X = i} in terms of P{X =i – 1}.
(b) Use part (a) to compute P{X = i} for i = 0, 1, 2, 3, 4, 5 when n = m = 10.
k = 5, by starting with P{X = 0}.
(c) Based on the recursion in part (a), write a program to compute the hyper-
%3D
%3D
10,
%3D
%3D
%3D
geometric distribution function.
(d) Use your program from part (c) to compute P{X < 10} when n = m= 30,
k = 15.
20. Independent trials, each of which is a success with probability p, are successively
performed. Let X denote the first trial resulting in a success. That is, X will equal
k if the first k-1 trials are all failures and the kth a success. X is called a geometric
random variable. Compute
(a) P{X= k}, k = 1, 2, ...;
(b) E[X].
%3D
Let Y denote the number of trials needed to obtain r successes. Y is called a
negative binomial random variable. Compute
(c) P(Y = k}, k= r,r+1,....
(Hint: In order for Y to equal k, how many successes must result in the first k:
trials and what must be the outcome of trial k?)
(d) Show that
E[Y] = rlp
(Hint: Write Y = Y1++ Y, where Y; is the number of trials needed to go
from a total of i-1 to a total of i successes.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc75d2948-de5d-4252-902e-2fb229e41c51%2Ff34abe7d-d186-4d4f-936b-8124e933dbcd%2Fzt86vhh.jpeg&w=3840&q=75)
Transcribed Image Text:Chapter 5: Special Random Variables
17. IfX is a Poisson random variable with mean 2, show that P{X=i} first increases
and then decreases as i increases, reaching its maximum value when i is the largest
000
18. A contractor purchases a shipment of 100 transistors. It is his policy to test 10
of these transistors and to keep the shipment only if at least 9 of the
working condition. If the shipment contains 20 defective transistors, what is the
integer less than or equal to 2.
probability it will be kept?
19. Let X denote a hypergeometric random variable with parameters n, m, and k.
That is,
()C)
(w)
i = 0, 1,..., min(k, n)
P{X = i} =
%3D
(a) Derive a formula for P{X = i} in terms of P{X =i – 1}.
(b) Use part (a) to compute P{X = i} for i = 0, 1, 2, 3, 4, 5 when n = m = 10.
k = 5, by starting with P{X = 0}.
(c) Based on the recursion in part (a), write a program to compute the hyper-
%3D
%3D
10,
%3D
%3D
%3D
geometric distribution function.
(d) Use your program from part (c) to compute P{X < 10} when n = m= 30,
k = 15.
20. Independent trials, each of which is a success with probability p, are successively
performed. Let X denote the first trial resulting in a success. That is, X will equal
k if the first k-1 trials are all failures and the kth a success. X is called a geometric
random variable. Compute
(a) P{X= k}, k = 1, 2, ...;
(b) E[X].
%3D
Let Y denote the number of trials needed to obtain r successes. Y is called a
negative binomial random variable. Compute
(c) P(Y = k}, k= r,r+1,....
(Hint: In order for Y to equal k, how many successes must result in the first k:
trials and what must be the outcome of trial k?)
(d) Show that
E[Y] = rlp
(Hint: Write Y = Y1++ Y, where Y; is the number of trials needed to go
from a total of i-1 to a total of i successes.)
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