n/2 for some positive integer n and B 6.46 Suppose that Y has a gamma distribution with a = equal to some specified value. Use the method of moment-generating functions to show that W = 2Y/B has a x² distribution with n degrees of freedom. nich 35 and 6 42 Use the result in
n/2 for some positive integer n and B 6.46 Suppose that Y has a gamma distribution with a = equal to some specified value. Use the method of moment-generating functions to show that W = 2Y/B has a x² distribution with n degrees of freedom. nich 35 and 6 42 Use the result in
n/2 for some positive integer n and B 6.46 Suppose that Y has a gamma distribution with a = equal to some specified value. Use the method of moment-generating functions to show that W = 2Y/B has a x² distribution with n degrees of freedom. nich 35 and 6 42 Use the result in
Please do number 6.46 and show work for finding the moment generating function
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6.41
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6.47
function of U = Y² + Y²/².
Suppose that Y₁ and Y2 are independent, standard normal random variables. Find the density
Let Y₁, Y2, ..., Y, be independent, normal random variables, each with mean and variance o².
U = Σi=1 a¡Y₁.
Let a₁, a2, ..., an denote known constants. Find the density function of the linear combination
A type of elevator has a maximum weight capacity Y₁, which is normally distributed with mean
5000 pounds and standard deviation 300 pounds. For a certain building equipped with this type
of elevator, the elevator's load, Y2, is a normally distributed random variable with mean 4000
pounds and standard deviation 400 pounds. For any given time that the elevator is in use, find
the probability that it will be overloaded, assuming that Y₁ and Y₂ are independent.
mean and variance o2.
Refer to Exercise 6.41. Let Y₁, Y₂, ..., Y,, be independent, normal random variables, each with
a
Find the density function of Y =
Exercises 323
-
=
n
n
ΣΥ
i=1
Y₁.
16 and n =
b If o²
25, what is the probability that the sample mean, Y, takes on a value
that is within one unit of the population mean, u? That is, find P(Y- | ≤ 1).
c If o² = 16, find P(Y - μ| ≤ 1) if n = 36, n = 64, and n = 81. Interpret the results of
your calculations.
The weight (in pounds) of "medium-size" watermelons is normally distributed with mean 15
and variance 4. A packing container for several melons has a nominal capacity of 140 pounds.
What is the maximum number of melons that should be placed in a single packing container
if the nominal weight limit is to be exceeded only 5% of the time? Give reasons for your
answer.
The manager of a construction job needs to figure prices carefully before submitting a bid. He
also needs to account for uncertainty (variability) in the amounts of products he might need.
To oversimplify the real situation, suppose that a project manager treats the amount of sand, in
yards, needed for a construction project as a random variable Y₁, which is normally distributed
with mean 10 yards and standard deviation .5 yard. The amount of cement mix needed, in
hundreds of pounds, is a random variable Y2, which is normally distributed with mean 4 and
standard deviation .2. The sand costs $7 per yard, and the cement mix costs $3 per hundred
pounds. Adding $100 for other costs, he computes his total cost to be
U = 100+7Y₁ +3Y₂.
If Y₁ and Y₂ are independent, how much should the manager bid to ensure that the true costs
will exceed the amount bid with a probability of only .01? Is the independence assumption
reasonable here?
Suppose that Y has a gamma distribution with a = n/2 for some positive integer n and ß
equal to some specified value. Use the method of moment-generating functions to show that
W = 2Y/B has a x² distribution with n degrees of freedom.
A random variable Y has a gamma distribution with a = 3.5 and ß = 4.2. Use the result in
Exercise 6.46 and the percentage points for the x² distributions given in Table 6, Appendix 3,
to find P(Y> 33.627).
6.48 In a missile-testing program, one random variable of interest is the distance between the point
at which the missile lands and the center of the target at which the missile was aimed. If we
think of the center of the target as the origin of a coordinate system, we can let Y₁ denote
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
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