*4.186 The random variable Y, with a density function given by mym-1 α -e-ym/a₁ 0 ≤ y ≤co, a, m > 0 is said to have a Weibull distribution. The Weibull density function provides a good model for the distribution of length of life for many mechanical devices and biological plants and animals. Find the mean and variance for a Weibull distributed random variable with m = 2. *4.187 Refer to Exercise 4.186. Resistors used in the construction of an aircraft guidance system have life lengths that follow a Weibull distribution with m = 2 and a = 10 (with measurements in thousands of hours). f(y) = a Find the probability that the life length of a randomly selected resistor of this type exceeds 5000 hours. b If three resistors of this type are operating independently, find the probability that exactly one of the three will burn out prior to 5000 hours of use.
*4.186 The random variable Y, with a density function given by mym-1 α -e-ym/a₁ 0 ≤ y ≤co, a, m > 0 is said to have a Weibull distribution. The Weibull density function provides a good model for the distribution of length of life for many mechanical devices and biological plants and animals. Find the mean and variance for a Weibull distributed random variable with m = 2. *4.187 Refer to Exercise 4.186. Resistors used in the construction of an aircraft guidance system have life lengths that follow a Weibull distribution with m = 2 and a = 10 (with measurements in thousands of hours). f(y) = a Find the probability that the life length of a randomly selected resistor of this type exceeds 5000 hours. b If three resistors of this type are operating independently, find the probability that exactly one of the three will burn out prior to 5000 hours of use.
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Please how do I solve 4.187?
![pomal
*4.186
*4.187
*4.188
*4.189
*4.190
a
a
200 udel
b Suppose that Y₁ is a random variable with density function fi (y) and that E(Y₁) = ₁ and
Var(Y₁) = 0; and similarly suppose that Y₂ is a random variable with density function
f2(y) and that E(Y₂) = ₂ and Var(Y₂) = o. Assume that Y is a random variable whose
density is a mixture of the densities corresponding to Y₁ and Y₂. Show that
E(Y) = αμ + (1 – a)μ2·
Var(Y) = ao? + (1-a)o2+a(1-a)[μ₁ - ₂1².
[Hint: E(Y) = μ+o?, i = 1,2.] togel od
The random variable Y, with a density function given by
mym-1
f(y) =
e-ym la
0 ≤ y ≤ ∞o, a, m > 0
is said to have a Weibull distribution. The Weibull density function provides a good model
for the distribution of length of life for many mechanical devices and biological plants and
animals. Find the mean and variance for a Weibull distributed random variable with m = 2.
Show that f(y) is a density function. Such a density function is often referred to as a
mixture of two density functions.
i
ii
Refer to Exercise 4.186. Resistors used in the construction of an aircraft guidance system have
life lengths that follow a Weibull distribution with m = 2 and a = 10 (with measurements in
thousands of hours).
din Supplementary Exercises 219
a
b
Find the probability that the life length of a randomly selected resistor of this type exceeds
5000 hours.
b If three resistors of this type are operating independently, find the probability that exactly
one of the three will burn out prior to 5000 hours of use.
Refer to Exercise 4.186.
a What is the usual name of the distribution of a random variable that has a Weibull distri-
bution with m = 1?
b Derive, in terms of the parameters a and m, the mean and variance of a Weibull distributed
random variable.
If n > 2 is an integer, the distribution with density given by
1
B(1/2, [n-2]/2)
0,
elsewhere.
is called the r distribution. Derive the mean and variance of a random variable with the r
distribution.
f(y) =
A function sometimes associated with continuous nonnegative random variables is the failure
rate (or hazard rate) function, which is defined by
r(t) =
-(1- y²)(n-4)/2, -1≤ y ≤ 1,
f (t)
1- F(t)
for a density function f(t) with corresponding distribution function F(t). If we think of the
random variable in question as being the length of life of a component, r(t) is proportional to
the probability of failure in a small interval after t, given that the component has survived up
to time t. Show that,
for an exponential density function, r(t) is constant.
for a Weibull density function with m > 1, r(t) is an increasing function of t. (See Exercise
4186)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22752cce-aae9-4c92-9942-5544ae988fbe%2F82093e83-70fd-4b24-a8e6-ba29a2d97ef4%2Fcqxio5o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:pomal
*4.186
*4.187
*4.188
*4.189
*4.190
a
a
200 udel
b Suppose that Y₁ is a random variable with density function fi (y) and that E(Y₁) = ₁ and
Var(Y₁) = 0; and similarly suppose that Y₂ is a random variable with density function
f2(y) and that E(Y₂) = ₂ and Var(Y₂) = o. Assume that Y is a random variable whose
density is a mixture of the densities corresponding to Y₁ and Y₂. Show that
E(Y) = αμ + (1 – a)μ2·
Var(Y) = ao? + (1-a)o2+a(1-a)[μ₁ - ₂1².
[Hint: E(Y) = μ+o?, i = 1,2.] togel od
The random variable Y, with a density function given by
mym-1
f(y) =
e-ym la
0 ≤ y ≤ ∞o, a, m > 0
is said to have a Weibull distribution. The Weibull density function provides a good model
for the distribution of length of life for many mechanical devices and biological plants and
animals. Find the mean and variance for a Weibull distributed random variable with m = 2.
Show that f(y) is a density function. Such a density function is often referred to as a
mixture of two density functions.
i
ii
Refer to Exercise 4.186. Resistors used in the construction of an aircraft guidance system have
life lengths that follow a Weibull distribution with m = 2 and a = 10 (with measurements in
thousands of hours).
din Supplementary Exercises 219
a
b
Find the probability that the life length of a randomly selected resistor of this type exceeds
5000 hours.
b If three resistors of this type are operating independently, find the probability that exactly
one of the three will burn out prior to 5000 hours of use.
Refer to Exercise 4.186.
a What is the usual name of the distribution of a random variable that has a Weibull distri-
bution with m = 1?
b Derive, in terms of the parameters a and m, the mean and variance of a Weibull distributed
random variable.
If n > 2 is an integer, the distribution with density given by
1
B(1/2, [n-2]/2)
0,
elsewhere.
is called the r distribution. Derive the mean and variance of a random variable with the r
distribution.
f(y) =
A function sometimes associated with continuous nonnegative random variables is the failure
rate (or hazard rate) function, which is defined by
r(t) =
-(1- y²)(n-4)/2, -1≤ y ≤ 1,
f (t)
1- F(t)
for a density function f(t) with corresponding distribution function F(t). If we think of the
random variable in question as being the length of life of a component, r(t) is proportional to
the probability of failure in a small interval after t, given that the component has survived up
to time t. Show that,
for an exponential density function, r(t) is constant.
for a Weibull density function with m > 1, r(t) is an increasing function of t. (See Exercise
4186)
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