6.54 Let Y₁, Y2,..., Y, be independent Poisson random variables with means 1, 2,..., An respectively. Find the a probability function of Y. b conditional probability function of Y₁, given that Y = m. Y₁ = m. c conditional probability function of Y₁+Y2, given that 6.55 Customers arrive at a department store checkout counter according to a Poisson distribution with a mean of 7 per hour. In a given two-hour period, what is the probability that 20 or more customers will arrive at the counter? 6.56 The length of time necessary to tune up a car is exponentially distributed with a mean of .5 hour. If two cars are waiting for a tune-up and the service times are independent, what is the probability that the total time for the two tune-ups will exceed 1.5 hours? [Hint: Recall the result of Example 6.12.] 6.57 Let Y, Y2,..., Y,, be independent random variables such that each Y, has a gamma distribution with parameters a, and B. That is, the distributions of the Y's might have different a's, but all have the same value for ẞ. Prove that U Y₁+Y₂++Y,, has a gamma distribution with parameters a₁+α₂++α, and B. 6.58 We saw in Exercise 5.159 that the negative binomial random variable Y can be written as Y=W, where W₁, W2,..., W, are independent geometric random variables with parameter p. 349/939 6.6 Multivariable Transformations Using Jacobians (Optional) a Use this fact to derive the moment-generating function for Y. b Use the moment-generating function to show that E(Y) = r/p and V(Y) = r(1 - p)/p². c Find the conditional probability function for W₁, given that Y = W₁+ W₂+...+W, = m. 6.59 Show that if Y, has a x² distribution with v₁ degrees of freedom and Y2 has a x² distribution with v2 degrees of freedom, then U = Y₁+ Y2 has a x2 distribution with v₁ + v2 degrees of freedom, provided that Y, and Y2 are independent. 6.60 Suppose that W = Y₁ + Y2 where Y₁ and Y2 are independent. If W has a x2 distribution with v degrees of freedom and W₁ has a x² distribution with v₁
6.54 Let Y₁, Y2,..., Y, be independent Poisson random variables with means 1, 2,..., An respectively. Find the a probability function of Y. b conditional probability function of Y₁, given that Y = m. Y₁ = m. c conditional probability function of Y₁+Y2, given that 6.55 Customers arrive at a department store checkout counter according to a Poisson distribution with a mean of 7 per hour. In a given two-hour period, what is the probability that 20 or more customers will arrive at the counter? 6.56 The length of time necessary to tune up a car is exponentially distributed with a mean of .5 hour. If two cars are waiting for a tune-up and the service times are independent, what is the probability that the total time for the two tune-ups will exceed 1.5 hours? [Hint: Recall the result of Example 6.12.] 6.57 Let Y, Y2,..., Y,, be independent random variables such that each Y, has a gamma distribution with parameters a, and B. That is, the distributions of the Y's might have different a's, but all have the same value for ẞ. Prove that U Y₁+Y₂++Y,, has a gamma distribution with parameters a₁+α₂++α, and B. 6.58 We saw in Exercise 5.159 that the negative binomial random variable Y can be written as Y=W, where W₁, W2,..., W, are independent geometric random variables with parameter p. 349/939 6.6 Multivariable Transformations Using Jacobians (Optional) a Use this fact to derive the moment-generating function for Y. b Use the moment-generating function to show that E(Y) = r/p and V(Y) = r(1 - p)/p². c Find the conditional probability function for W₁, given that Y = W₁+ W₂+...+W, = m. 6.59 Show that if Y, has a x² distribution with v₁ degrees of freedom and Y2 has a x² distribution with v2 degrees of freedom, then U = Y₁+ Y2 has a x2 distribution with v₁ + v2 degrees of freedom, provided that Y, and Y2 are independent. 6.60 Suppose that W = Y₁ + Y2 where Y₁ and Y2 are independent. If W has a x2 distribution with v degrees of freedom and W₁ has a x² distribution with v₁
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
Problem 11ECP: A manufacturer has determined that a machine averages one faulty unit for every 500 it produces....
Related questions
Question
![6.54 Let Y₁, Y2,..., Y, be independent Poisson random variables with means 1, 2,..., An
respectively. Find the
a probability function of
Y.
b conditional probability function of Y₁, given that Y = m.
Y₁ = m.
c conditional probability function of Y₁+Y2, given that
6.55 Customers arrive at a department store checkout counter according to a Poisson distribution
with a mean of 7 per hour. In a given two-hour period, what is the probability that 20 or more
customers will arrive at the counter?
6.56 The length of time necessary to tune up a car is exponentially distributed with a mean of
.5 hour. If two cars are waiting for a tune-up and the service times are independent, what is
the probability that the total time for the two tune-ups will exceed 1.5 hours? [Hint: Recall the
result of Example 6.12.]
6.57 Let Y, Y2,..., Y,, be independent random variables such that each Y, has a gamma distribution
with parameters a, and B. That is, the distributions of the Y's might have different a's, but all
have the same value for ẞ. Prove that U Y₁+Y₂++Y,, has a gamma distribution with
parameters a₁+α₂++α, and B.
6.58 We saw in Exercise 5.159 that the negative binomial random variable Y can be written as
Y=W, where W₁, W2,..., W, are independent geometric random variables with
parameter p.
349/939
6.6 Multivariable Transformations Using Jacobians (Optional)
a Use this fact to derive the moment-generating function for Y.
b Use the moment-generating function to show that E(Y) = r/p and V(Y) = r(1 - p)/p².
c Find the conditional probability function for W₁, given that Y = W₁+ W₂+...+W, = m.
6.59 Show that if Y, has a x² distribution with v₁ degrees of freedom and Y2 has a x² distribution
with v2 degrees of freedom, then U = Y₁+ Y2 has a x2 distribution with v₁ + v2 degrees of
freedom, provided that Y, and Y2 are independent.
6.60 Suppose that W = Y₁ + Y2 where Y₁ and Y2 are independent. If W has a x2 distribution with
v degrees of freedom and W₁ has a x² distribution with v₁ <v degrees of freedom, show that
Y₂ has a x2 distribution with v-v₁ degrees of freedom.
6.61
Refer to Exercise 6.52. Suppose that W = Y₁ + Y2 where Y₁ and Y2 are independent. If W has
a Poisson distribution with mean and W₁ has a Poisson distribution with mean ₁ <λ, show
that Y2 has a Poisson distribution with mean λ - λ₁.
*6.62 Let Y₁ and Y2 be independent normal random variables, each with mean 0 and variance o².
Define U₁ = Y₁+Y2 and U₂ = Y₁- Y2. Show that U₁ and U2 are independent normal random
variables, each with mean 0 and variance 202. [Hint: If (U1, U2) has a joint moment-generating
function m(11, 12), then U₁ and U2 are independent if and only if m(11, 12) = mu,(t)mu₂ (12).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f89dc84-5360-495d-a77a-10dc4ac0ca49%2Fbf301e8c-5df2-4371-8391-52742257ac85%2F0l1ctns_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6.54 Let Y₁, Y2,..., Y, be independent Poisson random variables with means 1, 2,..., An
respectively. Find the
a probability function of
Y.
b conditional probability function of Y₁, given that Y = m.
Y₁ = m.
c conditional probability function of Y₁+Y2, given that
6.55 Customers arrive at a department store checkout counter according to a Poisson distribution
with a mean of 7 per hour. In a given two-hour period, what is the probability that 20 or more
customers will arrive at the counter?
6.56 The length of time necessary to tune up a car is exponentially distributed with a mean of
.5 hour. If two cars are waiting for a tune-up and the service times are independent, what is
the probability that the total time for the two tune-ups will exceed 1.5 hours? [Hint: Recall the
result of Example 6.12.]
6.57 Let Y, Y2,..., Y,, be independent random variables such that each Y, has a gamma distribution
with parameters a, and B. That is, the distributions of the Y's might have different a's, but all
have the same value for ẞ. Prove that U Y₁+Y₂++Y,, has a gamma distribution with
parameters a₁+α₂++α, and B.
6.58 We saw in Exercise 5.159 that the negative binomial random variable Y can be written as
Y=W, where W₁, W2,..., W, are independent geometric random variables with
parameter p.
349/939
6.6 Multivariable Transformations Using Jacobians (Optional)
a Use this fact to derive the moment-generating function for Y.
b Use the moment-generating function to show that E(Y) = r/p and V(Y) = r(1 - p)/p².
c Find the conditional probability function for W₁, given that Y = W₁+ W₂+...+W, = m.
6.59 Show that if Y, has a x² distribution with v₁ degrees of freedom and Y2 has a x² distribution
with v2 degrees of freedom, then U = Y₁+ Y2 has a x2 distribution with v₁ + v2 degrees of
freedom, provided that Y, and Y2 are independent.
6.60 Suppose that W = Y₁ + Y2 where Y₁ and Y2 are independent. If W has a x2 distribution with
v degrees of freedom and W₁ has a x² distribution with v₁ <v degrees of freedom, show that
Y₂ has a x2 distribution with v-v₁ degrees of freedom.
6.61
Refer to Exercise 6.52. Suppose that W = Y₁ + Y2 where Y₁ and Y2 are independent. If W has
a Poisson distribution with mean and W₁ has a Poisson distribution with mean ₁ <λ, show
that Y2 has a Poisson distribution with mean λ - λ₁.
*6.62 Let Y₁ and Y2 be independent normal random variables, each with mean 0 and variance o².
Define U₁ = Y₁+Y2 and U₂ = Y₁- Y2. Show that U₁ and U2 are independent normal random
variables, each with mean 0 and variance 202. [Hint: If (U1, U2) has a joint moment-generating
function m(11, 12), then U₁ and U2 are independent if and only if m(11, 12) = mu,(t)mu₂ (12).]
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