29. Let I = f e2 dx. (a) Show that for any u and o 1 8. V2Ta / e-(x-120² dx = 1 is equivalent to I = 27. (b) Show that I = /2n by writing -312 %3D e-(x²+y³)/2 dy 3 dx dy 212 dx 0- and then evaluating the double integral by means of a change of variables to polar coordinates. (That is, let x=r cos 0, y = r sin 0, dx dy = r dr de 30. A random variable X is said to have a lognormal distribution if log X is normally distributed. If X is lognormal with E[log X] = µ and Var(log X) = o², deter- mine the distribution function of X. That is, what is P{X < x}? 31. The salaries of pediatric physicians are approximately normally distributed. If 25 percent of these physicians earn below 180, 000 and 25 percent earn above 320, 000, what fraction earn (a) below 250, 000; (b) between 260, 00 and 300, 000? 32. The sample mean and sample standard deviation on your economics examina- tion were 60 and 20, respectively; the sample mean and sample standard devia- tion on your statistics examination were 55 and 10, respectively. You scored 70 on the economics exam and 62 on the statistics exam. Assuming that the two histograms of test scores are approximately normal histograms, (a) on which exam was your percentile score highest? (b) approximate the percentage of the scores on the economics exam that were below your sCore. (c) approximate the percentage of the scores on the statistics exam that were below your score. 33. Value at risk (VAR) has become a key concept in financial calculations. The VAR an investment is defined as that value v such that there is only a 1 percent chance that the loss from the investment will exceed v. of (a) If the gain from an investment is a normal random variable with mean and variance 49, determine the value at risk. (If X is the gain, then -X is the loss.) (b) Among a set of investments whose gains are all normally distributed show that the one having the smallest VAR is the one having the largest val ra mee with the groding system for Problems 203 u - 2.330, where u and o are the mean and variance of the gain from the .2 investment. 34. The annual rainfall in Cincinnati is normally distributed with mean 40.14 inches and standard deviation 8.7 inches. (a) What is the probability this year's rainfall will exceed 42 inches? (b) What is the probability that the sum of the next 2 years' rainfall will exceed 84 inches? (c) What is the probability that the sum of the next 3 years' rainfall will exceed 126 inches? (d) For parts (b) and (c), what independence assumptions are you making? 35. The height of adult women in the United States is normally distributed with mean 64.5 inches and standard deviation 2.4 inches. Find the probability that a randomly chosen woman is (a) less than 63 inches tall; (b) less than 70 inches tall; (c) between 63 and 70 inches tall. (d) Alice is 72 inches tall. What percentage of women is shorter than Alice? (e) Find the probability that the average of the heights of two randomly chosen women exceeds 66 inches. (f) Repeat part (e) for four randomly chosen women. 36. An IQ test produces scores that are normally distributed with mean value 100 and standard deviation 14.2. The top 1 percent of all scores are in what range? 37. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter A = 1. %3D (a) What is the probability that a repair time exceeds 2 hours? (b) What is the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2 hours? 38. The number of years a radio functions is exponentially distributed with 3: If Jones buys a used radio, what is the probability that it will be param- eter 2 working after an additional 10 years? 39. Jones figures that the total number of thousands of miles that a used auto can be driven before it would need to be junked is an exponential random variable with parameter 20. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed but rather is (in thousands of miles) uniformly distributed over (0, 40).
29. Let I = f e2 dx. (a) Show that for any u and o 1 8. V2Ta / e-(x-120² dx = 1 is equivalent to I = 27. (b) Show that I = /2n by writing -312 %3D e-(x²+y³)/2 dy 3 dx dy 212 dx 0- and then evaluating the double integral by means of a change of variables to polar coordinates. (That is, let x=r cos 0, y = r sin 0, dx dy = r dr de 30. A random variable X is said to have a lognormal distribution if log X is normally distributed. If X is lognormal with E[log X] = µ and Var(log X) = o², deter- mine the distribution function of X. That is, what is P{X < x}? 31. The salaries of pediatric physicians are approximately normally distributed. If 25 percent of these physicians earn below 180, 000 and 25 percent earn above 320, 000, what fraction earn (a) below 250, 000; (b) between 260, 00 and 300, 000? 32. The sample mean and sample standard deviation on your economics examina- tion were 60 and 20, respectively; the sample mean and sample standard devia- tion on your statistics examination were 55 and 10, respectively. You scored 70 on the economics exam and 62 on the statistics exam. Assuming that the two histograms of test scores are approximately normal histograms, (a) on which exam was your percentile score highest? (b) approximate the percentage of the scores on the economics exam that were below your sCore. (c) approximate the percentage of the scores on the statistics exam that were below your score. 33. Value at risk (VAR) has become a key concept in financial calculations. The VAR an investment is defined as that value v such that there is only a 1 percent chance that the loss from the investment will exceed v. of (a) If the gain from an investment is a normal random variable with mean and variance 49, determine the value at risk. (If X is the gain, then -X is the loss.) (b) Among a set of investments whose gains are all normally distributed show that the one having the smallest VAR is the one having the largest val ra mee with the groding system for Problems 203 u - 2.330, where u and o are the mean and variance of the gain from the .2 investment. 34. The annual rainfall in Cincinnati is normally distributed with mean 40.14 inches and standard deviation 8.7 inches. (a) What is the probability this year's rainfall will exceed 42 inches? (b) What is the probability that the sum of the next 2 years' rainfall will exceed 84 inches? (c) What is the probability that the sum of the next 3 years' rainfall will exceed 126 inches? (d) For parts (b) and (c), what independence assumptions are you making? 35. The height of adult women in the United States is normally distributed with mean 64.5 inches and standard deviation 2.4 inches. Find the probability that a randomly chosen woman is (a) less than 63 inches tall; (b) less than 70 inches tall; (c) between 63 and 70 inches tall. (d) Alice is 72 inches tall. What percentage of women is shorter than Alice? (e) Find the probability that the average of the heights of two randomly chosen women exceeds 66 inches. (f) Repeat part (e) for four randomly chosen women. 36. An IQ test produces scores that are normally distributed with mean value 100 and standard deviation 14.2. The top 1 percent of all scores are in what range? 37. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter A = 1. %3D (a) What is the probability that a repair time exceeds 2 hours? (b) What is the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2 hours? 38. The number of years a radio functions is exponentially distributed with 3: If Jones buys a used radio, what is the probability that it will be param- eter 2 working after an additional 10 years? 39. Jones figures that the total number of thousands of miles that a used auto can be driven before it would need to be junked is an exponential random variable with parameter 20. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed but rather is (in thousands of miles) uniformly distributed over (0, 40).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Concept explainers
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Topic Video
Question
100%
Question 33
![29. Let I = f e2 dx.
(a) Show that for any u and o
1
8.
V2Ta / e-(x-120² dx = 1
is equivalent to I = 27.
(b) Show that I = /2n by writing
-312
%3D
e-(x²+y³)/2
dy 3
dx dy
212 dx
0-
and then evaluating the double integral by means of a change of variables
to polar coordinates. (That is, let x=r cos 0, y = r sin 0, dx dy = r dr de
30. A random variable X is said to have a lognormal distribution if log X is normally
distributed. If X is lognormal with E[log X] = µ and Var(log X) = o², deter-
mine the distribution function of X. That is, what is P{X < x}?
31. The salaries of pediatric physicians are approximately normally distributed. If
25 percent of these physicians earn below 180, 000 and 25 percent earn above
320, 000, what fraction earn
(a) below 250, 000;
(b) between 260, 00 and 300, 000?
32. The sample mean and sample standard deviation on your economics examina-
tion were 60 and 20, respectively; the sample mean and sample standard devia-
tion on your statistics examination were 55 and 10, respectively. You scored 70
on the economics exam and 62 on the statistics exam. Assuming that the two
histograms of test scores are approximately normal histograms,
(a) on which exam was your percentile score highest?
(b) approximate the percentage of the scores on the economics exam that were
below your sCore.
(c) approximate the percentage of the scores on the statistics exam that were
below your score.
33. Value at risk (VAR) has become a key concept in financial calculations. The VAR
an investment is defined as that value v such that there is only a 1 percent
chance that the loss from the investment will exceed v.
of
(a) If the gain from an investment is a normal random variable with mean
and variance 49, determine the value at risk. (If X is the gain, then -X is
the loss.)
(b) Among a set of investments whose gains are all normally distributed show
that the one having the smallest VAR is the one having the largest val](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc75d2948-de5d-4252-902e-2fb229e41c51%2Fc2978924-1fa1-4b02-b07b-f2d3ce85988b%2F27ew2wl.jpeg&w=3840&q=75)
Transcribed Image Text:29. Let I = f e2 dx.
(a) Show that for any u and o
1
8.
V2Ta / e-(x-120² dx = 1
is equivalent to I = 27.
(b) Show that I = /2n by writing
-312
%3D
e-(x²+y³)/2
dy 3
dx dy
212 dx
0-
and then evaluating the double integral by means of a change of variables
to polar coordinates. (That is, let x=r cos 0, y = r sin 0, dx dy = r dr de
30. A random variable X is said to have a lognormal distribution if log X is normally
distributed. If X is lognormal with E[log X] = µ and Var(log X) = o², deter-
mine the distribution function of X. That is, what is P{X < x}?
31. The salaries of pediatric physicians are approximately normally distributed. If
25 percent of these physicians earn below 180, 000 and 25 percent earn above
320, 000, what fraction earn
(a) below 250, 000;
(b) between 260, 00 and 300, 000?
32. The sample mean and sample standard deviation on your economics examina-
tion were 60 and 20, respectively; the sample mean and sample standard devia-
tion on your statistics examination were 55 and 10, respectively. You scored 70
on the economics exam and 62 on the statistics exam. Assuming that the two
histograms of test scores are approximately normal histograms,
(a) on which exam was your percentile score highest?
(b) approximate the percentage of the scores on the economics exam that were
below your sCore.
(c) approximate the percentage of the scores on the statistics exam that were
below your score.
33. Value at risk (VAR) has become a key concept in financial calculations. The VAR
an investment is defined as that value v such that there is only a 1 percent
chance that the loss from the investment will exceed v.
of
(a) If the gain from an investment is a normal random variable with mean
and variance 49, determine the value at risk. (If X is the gain, then -X is
the loss.)
(b) Among a set of investments whose gains are all normally distributed show
that the one having the smallest VAR is the one having the largest val
Transcribed Image Text:ra mee with the
groding system
for
Problems
203
u - 2.330, where u and o are the mean and variance of the gain from the
.2
investment.
34. The annual rainfall in Cincinnati is normally distributed with mean 40.14 inches
and standard deviation 8.7 inches.
(a) What is the probability this year's rainfall will exceed 42 inches?
(b) What is the probability that the sum of the next 2 years' rainfall will exceed
84 inches?
(c) What is the probability that the sum of the next 3 years' rainfall will exceed
126 inches?
(d) For parts (b) and (c), what independence assumptions are you making?
35. The height of adult women in the United States is normally distributed with
mean 64.5 inches and standard deviation 2.4 inches. Find the probability that
a randomly chosen woman is
(a) less than 63 inches tall;
(b) less than 70 inches tall;
(c) between 63 and 70 inches tall.
(d) Alice is 72 inches tall. What percentage of women is shorter than Alice?
(e) Find the probability that the average of the heights of two randomly chosen
women exceeds 66 inches.
(f) Repeat part (e) for four randomly chosen women.
36. An IQ test produces scores that are normally distributed with mean value 100
and standard deviation 14.2. The top 1 percent of all scores are in what range?
37. The time (in hours) required to repair a machine is an exponentially distributed
random variable with parameter A = 1.
%3D
(a) What is the probability that a repair time exceeds 2 hours?
(b) What is the conditional probability that a repair takes at least 3 hours, given
that its duration exceeds 2 hours?
38. The number of years a radio functions is exponentially distributed with
3: If Jones buys a used radio, what is the probability that it will be
param-
eter 2
working after an additional 10 years?
39. Jones figures that the total number of thousands of miles that a used auto can be
driven before it would need to be junked is an exponential random variable with
parameter 20. Smith has a used car that he claims has been driven only 10,000
miles. If Jones purchases the car, what is the probability that she would get
at least 20,000 additional miles out of it? Repeat under the assumption that
the lifetime mileage of the car is not exponentially distributed but rather is (in
thousands of miles) uniformly distributed over (0, 40).
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 3 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman