150 INTRODUCTION TO PROBABILITY AND STATISTICS reasonable to assume that these chips exhibit a constant hazard rate. Let (a) In a practical sense, what are the main causes of failure of these chips hazard rate be given by p(t) = .02. (Time is in years.). (b) What is the reliability function for chips of this type? (c) What is the reliability of a chip 20 years after it has been put into use! (d) What is the failure density for these chips? (e) What type of random variable is X, the time to failure of a chip? (f) What is the mean and variance for X? (g) What is the probability that a chip will be operable for at least 30. 60. The random variable X, the time to failure (in thousands of miles driven a the signal lights on an automobile has a Weibull distribution with a = B = 2. (a) Find the density, mean, and variance for X. (b) Find the reliability function for X. (c) What is the reliability of these lights at 5000 miles? At 10,000 miles! %3D (d) What is the hazard rate function? (e) What is the hazard rate at 5000 miles? At 10,000 miles? (f) What is the probability that the lights will fail during the first 3000min driven? 61. Show that for a > 0 and B > 0, aßxB-le-ax dx= 1 %3D thereby showing that the nonnegative function given in Definition 4.7.1 density for a continuous random variable. Hint: Let z = ax". %3D 62. Let X be a Weibull random variable with parameters a and ß. Show th E[X²] = a¯2ßT°(1 + 2/B). Hint: In evaluating briw Iss |x?aBxB-le-ax® dr 0. B let z = æx³. Evaluate the integral in a manner similar to that used in the pr 63. Use the result of Exercise 62 to find Var X for a Weibull random variable m parameters a and ß, thus completing the proof of Theorem 4.7.1.
150 INTRODUCTION TO PROBABILITY AND STATISTICS reasonable to assume that these chips exhibit a constant hazard rate. Let (a) In a practical sense, what are the main causes of failure of these chips hazard rate be given by p(t) = .02. (Time is in years.). (b) What is the reliability function for chips of this type? (c) What is the reliability of a chip 20 years after it has been put into use! (d) What is the failure density for these chips? (e) What type of random variable is X, the time to failure of a chip? (f) What is the mean and variance for X? (g) What is the probability that a chip will be operable for at least 30. 60. The random variable X, the time to failure (in thousands of miles driven a the signal lights on an automobile has a Weibull distribution with a = B = 2. (a) Find the density, mean, and variance for X. (b) Find the reliability function for X. (c) What is the reliability of these lights at 5000 miles? At 10,000 miles! %3D (d) What is the hazard rate function? (e) What is the hazard rate at 5000 miles? At 10,000 miles? (f) What is the probability that the lights will fail during the first 3000min driven? 61. Show that for a > 0 and B > 0, aßxB-le-ax dx= 1 %3D thereby showing that the nonnegative function given in Definition 4.7.1 density for a continuous random variable. Hint: Let z = ax". %3D 62. Let X be a Weibull random variable with parameters a and ß. Show th E[X²] = a¯2ßT°(1 + 2/B). Hint: In evaluating briw Iss |x?aBxB-le-ax® dr 0. B let z = æx³. Evaluate the integral in a manner similar to that used in the pr 63. Use the result of Exercise 62 to find Var X for a Weibull random variable m parameters a and ß, thus completing the proof of Theorem 4.7.1.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
#60 please
![150 INTRODUCTION TO PROBABILITY AND STATISTICS
reasonable to assume that these chips exhibit a constant hazard rate. Let
(a) In a practical sense, what are the main causes of failure of these chips
hazard rate be given by p(t) = .02. (Time is in years.).
(b) What is the reliability function for chips of this type?
(c) What is the reliability of a chip 20 years after it has been put into use!
(d) What is the failure density for these chips?
(e) What type of random variable is X, the time to failure of a chip?
(f) What is the mean and variance for X?
(g) What is the probability that a chip will be operable for at least 30.
60. The random variable X, the time to failure (in thousands of miles driven a
the signal lights on an automobile has a Weibull distribution with a =
B = 2.
(a) Find the density, mean, and variance for X.
(b) Find the reliability function for X.
(c) What is the reliability of these lights at 5000 miles? At 10,000 miles!
%3D
(d) What is the hazard rate function?
(e) What is the hazard rate at 5000 miles? At 10,000 miles?
(f) What is the probability that the lights will fail during the first 3000min
driven?
61. Show that for a > 0 and B > 0,
aßxB-le-ax dx= 1
%3D
thereby showing that the nonnegative function given in Definition 4.7.1
density for a continuous random variable. Hint: Let z = ax".
%3D
62. Let X be a Weibull random variable with parameters a and ß. Show th
E[X²] = a¯2ßT°(1 + 2/B). Hint: In evaluating
briw Iss
|x?aBxB-le-ax® dr
0.
B let z = æx³. Evaluate the integral in a manner similar to that used in the pr
63. Use the result of Exercise 62 to find Var X for a Weibull random variable m
parameters a and ß, thus completing the proof of Theorem 4.7.1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F2fe8b76a-c3bf-49d2-8f03-2cadacc08f46%2F782spn8.jpeg&w=3840&q=75)
Transcribed Image Text:150 INTRODUCTION TO PROBABILITY AND STATISTICS
reasonable to assume that these chips exhibit a constant hazard rate. Let
(a) In a practical sense, what are the main causes of failure of these chips
hazard rate be given by p(t) = .02. (Time is in years.).
(b) What is the reliability function for chips of this type?
(c) What is the reliability of a chip 20 years after it has been put into use!
(d) What is the failure density for these chips?
(e) What type of random variable is X, the time to failure of a chip?
(f) What is the mean and variance for X?
(g) What is the probability that a chip will be operable for at least 30.
60. The random variable X, the time to failure (in thousands of miles driven a
the signal lights on an automobile has a Weibull distribution with a =
B = 2.
(a) Find the density, mean, and variance for X.
(b) Find the reliability function for X.
(c) What is the reliability of these lights at 5000 miles? At 10,000 miles!
%3D
(d) What is the hazard rate function?
(e) What is the hazard rate at 5000 miles? At 10,000 miles?
(f) What is the probability that the lights will fail during the first 3000min
driven?
61. Show that for a > 0 and B > 0,
aßxB-le-ax dx= 1
%3D
thereby showing that the nonnegative function given in Definition 4.7.1
density for a continuous random variable. Hint: Let z = ax".
%3D
62. Let X be a Weibull random variable with parameters a and ß. Show th
E[X²] = a¯2ßT°(1 + 2/B). Hint: In evaluating
briw Iss
|x?aBxB-le-ax® dr
0.
B let z = æx³. Evaluate the integral in a manner similar to that used in the pr
63. Use the result of Exercise 62 to find Var X for a Weibull random variable m
parameters a and ß, thus completing the proof of Theorem 4.7.1.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 6 steps with 14 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON