2. Iron bars produced by a machine can either be classified as "Good" with probability p, or "Bad" with probability q = 1- p. The quality of each iron bar produced is independent of each other. (a) Suppose there are n iron bars produced by the machine. Write down the probability that there are exactly r "Good" iron bars among the n of them for all positive integers r. 4 (b) Suppose the machine keeps producing iron bars until k "Good" ones are obtained. Let X be the total number of iron bars produced. Show that the probability mass function of X is given by px(x) = (x - 1)^(1-p)²-*, x = k₁k+1,... 1 (c) In a single day, the machine produces exactly N = n iron bars, where N has mean and variance given by E(N) = a, Var(N) = 3. Let W be the total number of "Good" iron bars produced in a single day. i. Find E(W) and Var(W) in terms of p, a and 3. You can use the mean and variance of any standard distributions without proof as long as you state them clearly. In ii. Determine the covariance between W and N in terms of p, a, and 8. Simplify your answer as much as possible. 1-
2. Iron bars produced by a machine can either be classified as "Good" with probability p, or "Bad" with probability q = 1- p. The quality of each iron bar produced is independent of each other. (a) Suppose there are n iron bars produced by the machine. Write down the probability that there are exactly r "Good" iron bars among the n of them for all positive integers r. 4 (b) Suppose the machine keeps producing iron bars until k "Good" ones are obtained. Let X be the total number of iron bars produced. Show that the probability mass function of X is given by px(x) = (x - 1)^(1-p)²-*, x = k₁k+1,... 1 (c) In a single day, the machine produces exactly N = n iron bars, where N has mean and variance given by E(N) = a, Var(N) = 3. Let W be the total number of "Good" iron bars produced in a single day. i. Find E(W) and Var(W) in terms of p, a and 3. You can use the mean and variance of any standard distributions without proof as long as you state them clearly. In ii. Determine the covariance between W and N in terms of p, a, and 8. Simplify your answer as much as possible. 1-
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Related questions
Question
![2. Iron bars produced by a machine can either be classified as "Good" with
probability p, or "Bad" with probability q = 1- p. The quality of each iron bar
produced is independent of each other.
(a) Suppose there are n iron bars produced by the machine. Write down the
probability that there are exactly r "Good" iron bars among the n of them for
all positive integers r.
(b) Suppose the machine keeps producing iron bars until k "Good" ones are
obtained. Let X be the total number of iron bars produced. Show that the
probability mass function of X is given by
Px(x) = (x − 1) ₁² (1 - p)² -*.
x=k, k +1,....
(c) In a single day, the machine produces exactly N = n iron bars, where N has
mean and variance given by
E(N) = a, Var(N) = 3.
Let W be the total number of "Good" iron bars produced in a single day.
i. Find E(W) and Var(W) in terms of p, a and 3. You can use the mean
and variance of any standard distributions without proof as long as you
state them clearly.
ii. Determine the covariance between W and N in terms of p, a, and 3.
Simplify your answer as much as possible.
1-
13:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6647abd2-e1f6-49ac-960c-dc3d8332f4f0%2Fc6354571-9f3a-44d1-8f81-5b6f3a943683%2Fyuyfyvl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Iron bars produced by a machine can either be classified as "Good" with
probability p, or "Bad" with probability q = 1- p. The quality of each iron bar
produced is independent of each other.
(a) Suppose there are n iron bars produced by the machine. Write down the
probability that there are exactly r "Good" iron bars among the n of them for
all positive integers r.
(b) Suppose the machine keeps producing iron bars until k "Good" ones are
obtained. Let X be the total number of iron bars produced. Show that the
probability mass function of X is given by
Px(x) = (x − 1) ₁² (1 - p)² -*.
x=k, k +1,....
(c) In a single day, the machine produces exactly N = n iron bars, where N has
mean and variance given by
E(N) = a, Var(N) = 3.
Let W be the total number of "Good" iron bars produced in a single day.
i. Find E(W) and Var(W) in terms of p, a and 3. You can use the mean
and variance of any standard distributions without proof as long as you
state them clearly.
ii. Determine the covariance between W and N in terms of p, a, and 3.
Simplify your answer as much as possible.
1-
13:
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