Mathematical Statistics with Applications
7th Edition
ISBN: 9781133384380
Author: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer
Publisher: Cengage Learning US
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Question
Chapter 4.3, Problem 34E
To determine
Prove that
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Let X and Y be jointly continuous
random variables with joint PDF
given by:
fx₁y (x,y) = = X (1+3y² ) / (0₁2) (X) I (01) (y)
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a. Find the conditional PDF of X given y
b. Find P ( ₁ < x < 1/2 | Y = £)
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c. Find the marginal PDF of X.
Let x and y be joint continuous random variable with joint pdf
f XY (x, y) = {?x + 1 , x, y ≥ 0, x + y < 10, ??ℎ??????1. Find the constant c.2. Find the marginal PDF’S fX (x) and fY (y)3. Find P(Y<2X2)
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fx(x) = 2aI(0,1)(x) and Y has a pdf given by fy(y) = 2(1– y)I(0,1)(y). Find the distribution of
X + Y.
Chapter 4 Solutions
Mathematical Statistics with Applications
Ch. 4.2 - Prob. 1ECh. 4.2 - A box contains five keys, only one of which will...Ch. 4.2 - A Bernoulli random variable is one that assumes...Ch. 4.2 - Let Y be a binomial random variable with n = 1 and...Ch. 4.2 - Suppose that Y is a random variable that takes on...Ch. 4.2 - Consider a random variable with a geometric...Ch. 4.2 - Let Y be a binomial random variable with n=10 and...Ch. 4.2 - Prob. 8ECh. 4.2 - A random variable Y has the following distribution...Ch. 4.2 - Refer to the density function given in Exercise...
Ch. 4.2 - Suppose that Y possesses the density function...Ch. 4.2 - Prob. 12ECh. 4.2 - A supplier of kerosene has a 150-gallon tank that...Ch. 4.2 - A gas station operates two pumps, each of which...Ch. 4.2 - As a measure of intelligence, mice are timed when...Ch. 4.2 - Let Y possess a density function...Ch. 4.2 - Prob. 17ECh. 4.2 - Prob. 18ECh. 4.2 - Prob. 19ECh. 4.3 - Prob. 20ECh. 4.3 - If, as in Exercise 4.17, Y has density function...Ch. 4.3 - Prob. 22ECh. 4.3 - Prob. 23ECh. 4.3 - If Y is a continuous random variable with density...Ch. 4.3 - Prob. 25ECh. 4.3 - If Y is a continuous random variable with mean ...Ch. 4.3 - Prob. 27ECh. 4.3 - Prob. 28ECh. 4.3 - Prob. 29ECh. 4.3 - The proportion of time Y that an industrial robot...Ch. 4.3 - Prob. 31ECh. 4.3 - Weekly CPU time used by an accounting firm has...Ch. 4.3 - The pH of water samples from a specific lake is a...Ch. 4.3 - Prob. 34ECh. 4.3 - If Y is a continuous random variable such that...Ch. 4.3 - Prob. 36ECh. 4.3 - Prob. 37ECh. 4.4 - Suppose that Y has a uniform distribution over the...Ch. 4.4 - If a parachutist lands at a random point on a line...Ch. 4.4 - Suppose that three parachutists operate...Ch. 4.4 - Prob. 41ECh. 4.4 - Prob. 42ECh. 4.4 - A circle of radius r has area A = r2. If a random...Ch. 4.4 - Prob. 44ECh. 4.4 - Upon studying low bids for shipping contracts, a...Ch. 4.4 - 4.45 Upon studying low bids for shipping...Ch. 4.4 - The failure of a circuit board interrupts work...Ch. 4.4 - If a point is randomly located in an interval (a,...Ch. 4.4 - Prob. 49ECh. 4.4 - Prob. 50ECh. 4.4 - The cycle time for trucks hauling concrete to a...Ch. 4.4 - Refer to Exercise 4.51. Find the mean and variance...Ch. 4.4 - Prob. 53ECh. 4.4 - Prob. 54ECh. 4.4 - Refer to Exercise 4.54. Suppose that measurement...Ch. 4.4 - Refer to Example 4.7. Find the conditional...Ch. 4.4 - Prob. 57ECh. 4.5 - Use Table 4, Appendix 3, to find the following...Ch. 4.5 - Prob. 59ECh. 4.5 - Prob. 60ECh. 4.5 - What is the median of a normally distributed...Ch. 4.5 - If Z is a standard normal random variable, what is...Ch. 4.5 - A company that manufactures and bottles apple...Ch. 4.5 - The weekly amount of money spent on maintenance...Ch. 4.5 - In Exercise 4.64, how much should be budgeted for...Ch. 4.5 - A machining operation produces bearings with...Ch. 4.5 - Prob. 67ECh. 4.5 - Prob. 68ECh. 4.5 - Refer to Exercise 4.68. If students possessing a...Ch. 4.5 - Refer to Exercise 4.68. Suppose that three...Ch. 4.5 - Wires manufactured for use in a computer system...Ch. 4.5 - Prob. 72ECh. 4.5 - The width of bolts of fabric is normally...Ch. 4.5 - A soft-drink machine can be regulated so that it...Ch. 4.5 - The machine described in Exercise 4.75 has...Ch. 4.5 - The SAT and ACT college entrance exams are taken...Ch. 4.5 - Show that the maximum value of the normal density...Ch. 4.5 - Show that the normal density with parameters and ...Ch. 4.5 - Assume that Y is normally distributed with mean ...Ch. 4.6 - a If 0, () is defined by ()=0y1eydy, show that...Ch. 4.6 - Use the results obtained in Exercise 4.81 to prove...Ch. 4.6 - The magnitude of earthquakes recorded in a region...Ch. 4.6 - If Y has an exponential distribution and P(Y 2) =...Ch. 4.6 - Refer to Exercise 4.88. Of the next ten...Ch. 4.6 - The operator of a pumping station has observed...Ch. 4.6 - The length of time Y necessary to complete a key...Ch. 4.6 - Historical evidence indicates that times between...Ch. 4.6 - One-hour carbon monoxide concentrations in air...Ch. 4.6 - Prob. 95ECh. 4.6 - Prob. 96ECh. 4.6 - Prob. 97ECh. 4.6 - Consider the plant of Exercise 4.97. How much of...Ch. 4.6 - If 0 and is a positive integer, the...Ch. 4.6 - Prob. 100ECh. 4.6 - Applet Exercise Refer to Exercise 4.88. Suppose...Ch. 4.6 - Prob. 102ECh. 4.6 - Explosive devices used in mining operations...Ch. 4.6 - The lifetime (in hours) Y of an electronic...Ch. 4.6 - Four-week summer rainfall totals in a section of...Ch. 4.6 - The response times on an online computer terminal...Ch. 4.6 - Refer to Exercise 4.106. a. Use Tchebysheffs...Ch. 4.6 - The weekly amount of downtime Y (in hours) for an...Ch. 4.6 - If Y has a probability density function given by...Ch. 4.6 - Suppose that Y has a gamma distribution with...Ch. 4.6 - Prob. 112ECh. 4.7 - Prob. 120ECh. 4.7 - Prob. 122ECh. 4.7 - The relative humidity Y, when measured at a...Ch. 4.7 - The percentage of impurities per batch in a...Ch. 4.7 - Prob. 125ECh. 4.7 - Suppose that a random variable Y has a probability...Ch. 4.7 - Verify that if Y has a beta distribution with = ...Ch. 4.7 - Prob. 128ECh. 4.7 - During an eight-hour shift, the proportion of time...Ch. 4.7 - Prob. 130ECh. 4.7 - Errors in measuring the time of arrival of a wave...Ch. 4.7 - Prob. 132ECh. 4.7 - Prob. 133ECh. 4.7 - Prob. 134ECh. 4.7 - Prob. 135ECh. 4.9 - Suppose that the waiting time for the first...Ch. 4.9 - Prob. 137ECh. 4.9 - Example 4.16 derives the moment-generating...Ch. 4.9 - The moment-generating function of a normally...Ch. 4.9 - Identify the distributions of the random variables...Ch. 4.9 - If 1 2, derive the moment-generating function of...Ch. 4.9 - Refer to Exercises 4.141 and 4.137. Suppose that Y...Ch. 4.9 - The moment-generating function for the gamma...Ch. 4.9 - Consider a random variable Y with density function...Ch. 4.9 - A random variable Y has the density function...Ch. 4.10 - A manufacturer of tires wants to advertise a...Ch. 4.10 - A machine used to fill cereal boxes dispenses, on...Ch. 4.10 - Find P(|Y | 2) for Exercise 4.16. Compare with...Ch. 4.10 - Find P(|Y | 2) for the uniform random variable....Ch. 4.10 - Prob. 150ECh. 4.10 - Prob. 151ECh. 4.10 - Refer to Exercise 4.109. Find an interval that...Ch. 4.10 - Refer to Exercise 4.129. Find an interval for...Ch. 4.11 - A builder of houses needs to order some supplies...Ch. 4.11 - Prob. 157ECh. 4.11 - Consider the nail-firing device of Example 4.15....Ch. 4.11 - Prob. 159ECh. 4 - Prob. 160SECh. 4 - Prob. 161SECh. 4 - Prob. 162SECh. 4 - Prob. 163SECh. 4 - The length of life of oil-drilling bits depends...Ch. 4 - Prob. 165SECh. 4 - Prob. 166SECh. 4 - Prob. 167SECh. 4 - Prob. 168SECh. 4 - An argument similar to that of Exercise 4.168 can...Ch. 4 - Prob. 170SECh. 4 - Suppose that customers arrive at a checkout...Ch. 4 - Prob. 172SECh. 4 - Prob. 173SECh. 4 - Prob. 174SECh. 4 - Prob. 175SECh. 4 - If Y has an exponential distribution with mean ,...Ch. 4 - Prob. 180SECh. 4 - Prob. 181SECh. 4 - Prob. 182SECh. 4 - Prob. 183SECh. 4 - Prob. 184SECh. 4 - Prob. 185SECh. 4 - Prob. 186SECh. 4 - Refer to Exercise 4.186. Resistors used in the...Ch. 4 - Prob. 188SECh. 4 - Prob. 189SECh. 4 - Prob. 190SECh. 4 - Prob. 191SECh. 4 - The velocities of gas particles can be modeled by...Ch. 4 - Because P(YyYc)=F(y)F(c)1F(c) has the properties...Ch. 4 - Prob. 194SECh. 4 - Prob. 195SECh. 4 - Prob. 196SECh. 4 - Prob. 197SECh. 4 - Prob. 198SECh. 4 - Prob. 199SECh. 4 - Prob. 200SE
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- Let X be a random variable with pdf f(x) = kx*,-1arrow_forwardEach front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable—X for the right tire and Y for the left tire, with joint pdf f(x, y) = K(x2 + y2) 19 ≤ x ≤ 29, 19 ≤ y ≤ 29 0 otherwise (a) Determine the conditional pdf of Y given that X = x. fY|X(y|x) = for 19 ≤ y ≤ 29 Determine the conditional pdf of X given that Y = y. fX|Y(x|y) = for 19 ≤ x ≤ 29 (b) If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure of at least 25 psi? (It is known that K = 3 350,600 . Round your answer to three decimal places.) Compare this to P(Y ≥ 25). (Round your answer to three decimal places.)P(Y ≥ 25) = (c) If the pressure in the right tire is found to be 22 psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire? (It is known that K = 3 350,600…arrow_forwardLet X be a (continuous) uniform random variable on the interval [0,1] and Y be an exponential random variable with parameter lambda. Let X and Y be independent. What is the PDF of Z = X + Y.arrow_forward4. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = 9x(x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be E[X] = √rg(x)dx, to be the expectation of XY. 0 with a similar definition for Y but g replaced by h and x replaced by y. We also define E[XY] = (0,00)x (0,00) 110,00)x (0,00) 29 (x, y) dedy (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y]. [2]arrow_forwardQ. 5 Let X be a continuous uniform random variable on [-a, a]. Find the pdf for Y = g(X), where 0, x b. g(x) = 3x,arrow_forwardLet X and Y be jointly continuous random variables with joint PDF fxx(x, y) x, y > 0, x + y < 1 сх + 1 otherwise 1. Show the range of (X,Y), RxY, in the a – y plane. - 2. Find the constant c. 3. Find the marginal PDFS fx(x) and fy(y).arrow_forwardLet X, and X; be independent exponential random variables having pdfs f, (x) = e, x > 0 and fr,(x2) = e"", I3 > 0. Given that Y, = X,/X and Y, = X, + X2. a. Find the joint pdf of Y, and Y2 b. Find the marginal pdf of Y1.arrow_forwardC, D, and Earrow_forwardSuppose x and y are continuous random variables with joint pdf f(x,y)= 4(x-xy) if 0<x<1 and 0<y<1 and zero otherwise. What is probability p[X <Y ]arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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