 5.1: The Rayleigh distribution from Example 5.1.7 has PDF f(x) = xex2/2 ...
 5.2: (a) Make up a PDF f, with an application for which that PDF would b...
 5.3: Let F be the CDF of a continuous r.v., and f = F0 be the PDF. (a) S...
 5.4: Let X be a continuous r.v. with CDF F and PDF f. (a) Find the condi...
 5.5: A circle with a random radius R Unif(0, 1) is generated. Let A be i...
 5.6: The 689599.7% rule gives approximate probabilities of a Normal r....
 5.7: Let F(x) = 2 sin1 px , for 0 1, and let F(x) = 0 for x 0 and F(x) =...
 5.8: The Beta distribution with parameters a = 3, b = 2 has PDF f(x) = 1...
 5.9: The Cauchy distribution has PDF f(x) = 1 (1 + x2) , for all x. (We ...
 5.10: Let U Unif(0, 8). (a) Find P(U 2 (0, 2) [ (3, 7)) without using cal...
 5.11: Let U be a Uniform r.v. on the interval (1, 1) (be careful about mi...
 5.12: A stick is broken into two pieces, at a uniformly random break poin...
 5.13: A stick of length 1 is broken at a uniformly random point, yielding...
 5.14: Let U1,...,Un be i.i.d. Unif(0, 1), and X = max(U1,...,Un). What is...
 5.15: Let U Unif(0, 1). Using U, construct X Expo().
 5.16: Let U Unif(0, 1), and X = log U 1 U . Then X has the Logistic distr...
 5.17: Let U Unif(0, 1). As a function of U, create an r.v. X with CDF F(x...
 5.18: The Pareto distribution with parameter a > 0 has PDF f(x) = a/xa+1 ...
 5.19: Let F be a CDF which is continuous and strictly increasing. Let be ...
 5.20: Let X be a nonnegative r.v. with a continuous, strictly increasing ...
 5.21: Let Z N (0, 1). Create an r.v. Y N (1, 4), as a simplelooking func...
 5.22: Engineers sometimes work with the error function erf(z) = 2 p Z z 0...
 5.23: (a) Find the points of inflection of the N (0, 1) PDF ', i.e., the ...
 5.24: The distance between two points needs to be measured, in meters. Th...
 5.25: Alice is trying to transmit to Bob the answer to a yesno question,...
 5.26: A woman is pregnant, with a due date of January 10, 2014. Of course...
 5.27: We will show in the next chapter that if X1 and X2 are independent ...
 5.28: Walter and Carl both often need to travel from Location A to Locati...
 5.29: Let Z N (0, 1). We know from the 689599.7% rule that there is a 6...
 5.30: Let Y N (, 1). Use the fact that P(Y  < 1.96) 0.95 to construct a...
 5.31: Let Y = X, with X N (, 2). This is a welldefined continuous r.v....
 5.32: Let Z N (0, 1) and let S be a random sign independent of Z, i.e., S...
 5.33: Let Z N (0, 1). Find E ((Z)) without using LOTUS, where is the CDF ...
 5.34: Let Z N (0, 1) and X = Z2. Then the distribution of X is called Chi...
 5.35: Let Z N (0, 1), with CDF . The PDF of Z2 is the function g given by...
 5.36: Let Z N (0, 1). A measuring device is used to observe Z, but the de...
 5.37: Let Z N (0, 1), and c be a nonnegative constant. Find E(max(Z c, 0)...
 5.38: A post oce has 2 clerks. Alice enters the post oce while 2 other cu...
 5.39: Three students are working independently on their probability homew...
 5.40: Let T be the time until a radioactive particle decays, and suppose ...
 5.41: Fred wants to sell his car, after moving back to Blissville (where ...
 5.42: (a) Fred visits Blotchville again. He finds that the city has insta...
 5.43: Fred and Gretchen are waiting at a bus stop in Blotchville. Two bus...
 5.44: Joe is waiting in continuous time for a book called The Winds of Wi...
 5.45: The Exponential is the analog of the Geometric in continuous time. ...
 5.46: The Laplace distribution has PDF f(x) = 1 2 e x for all real x. T...
 5.47: Emails arrive in an inbox according to a Poisson process with rate ...
 5.48: Let T be the lifetime of a certain person (how long that person liv...
 5.49: Let T be the lifetime of a person (or animal or gadget), with CDF F...
 5.50: Find E(X3) for X Expo(), using LOTUS and the fact that E(X)=1/ and ...
 5.51: The Gumbel distribution is the distribution of log X with X Expo(1)...
 5.52: Explain intuitively why P(X<Y ) = P(Y <X) if X and Y are i.i.d., Y ...
 5.53: Let X be an r.v. (discrete or continuous) such that 0 X 1 always ho...
 5.54: The Rayleigh distribution from Example 5.1.7 has PDF f(x) = xex2/2 ...
 5.55: Consider an experiment where we observe the value of a random varia...
 5.56: (a) Suppose that we have a list of the populations of every country...
 5.57: (a) Let X1, X2,... be independent N (0, 4) r.v.s., and let J be the...
 5.58: The unit circle {(x, y) : x2 +y2 = 1} is divided into three arcs by...
 5.59: As in Example 5.7.3, athletes compete one at a time at the high jum...
 5.60: Tyrion, Cersei, and n other guests arrive at a party at i.i.d. time...
 5.61: Let X1, X2,... be the annual rainfalls in Boston (measured in inche...
Solutions for Chapter 5: Continuous Random Variables
Full solutions for Introduction to Probability  1st Edition
ISBN: 9781466575578
Solutions for Chapter 5: Continuous Random Variables
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Introduction to Probability was written by and is associated to the ISBN: 9781466575578. Chapter 5: Continuous Random Variables includes 61 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Probability, edition: 1. Since 61 problems in chapter 5: Continuous Random Variables have been answered, more than 9992 students have viewed full stepbystep solutions from this chapter.

Alias
In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.

Alternative hypothesis
In statistical hypothesis testing, this is a hypothesis other than the one that is being tested. The alternative hypothesis contains feasible conditions, whereas the null hypothesis speciies conditions that are under test

Bayes’ estimator
An estimator for a parameter obtained from a Bayesian method that uses a prior distribution for the parameter along with the conditional distribution of the data given the parameter to obtain the posterior distribution of the parameter. The estimator is obtained from the posterior distribution.

Biased estimator
Unbiased estimator.

Bivariate normal distribution
The joint distribution of two normal random variables

Causeandeffect diagram
A chart used to organize the various potential causes of a problem. Also called a ishbone diagram.

Chisquare (or chisquared) random variable
A continuous random variable that results from the sum of squares of independent standard normal random variables. It is a special case of a gamma random variable.

Chisquare test
Any test of signiicance based on the chisquare distribution. The most common chisquare tests are (1) testing hypotheses about the variance or standard deviation of a normal distribution and (2) testing goodness of it of a theoretical distribution to sample data

Combination.
A subset selected without replacement from a set used to determine the number of outcomes in events and sample spaces.

Components of variance
The individual components of the total variance that are attributable to speciic sources. This usually refers to the individual variance components arising from a random or mixed model analysis of variance.

Conditional mean
The mean of the conditional probability distribution of a random variable.

Conditional probability density function
The probability density function of the conditional probability distribution of a continuous random variable.

Control limits
See Control chart.

Covariance matrix
A square matrix that contains the variances and covariances among a set of random variables, say, X1 , X X 2 k , , … . The main diagonal elements of the matrix are the variances of the random variables and the offdiagonal elements are the covariances between Xi and Xj . Also called the variancecovariance matrix. When the random variables are standardized to have unit variances, the covariance matrix becomes the correlation matrix.

Decision interval
A parameter in a tabular CUSUM algorithm that is determined from a tradeoff between false alarms and the detection of assignable causes.

Defectsperunit control chart
See U chart

Eficiency
A concept in parameter estimation that uses the variances of different estimators; essentially, an estimator is more eficient than another estimator if it has smaller variance. When estimators are biased, the concept requires modiication.

Estimate (or point estimate)
The numerical value of a point estimator.

Fisher’s least signiicant difference (LSD) method
A series of pairwise hypothesis tests of treatment means in an experiment to determine which means differ.

Gaussian distribution
Another name for the normal distribution, based on the strong connection of Karl F. Gauss to the normal distribution; often used in physics and electrical engineering applications