Probability and Statistical Inference (9th Edition)
9th Edition
ISBN: 9780321923271
Author: Robert V. Hogg, Elliot Tanis, Dale Zimmerman
Publisher: PEARSON
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Chapter 3.1, Problem 1E
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QUESTION 18 - 1 POINT
Jessie is playing a dice game and bets $9 on her first roll. If a 10, 7, or 4 is rolled, she wins $9. This happens with a probability of . If an 8 or 2 is rolled, she loses her $9. This has a probability of J. If any other number is rolled, she does not win or lose, and the game continues. Find the expected value for Jessie on her first roll.
Round to the nearest cent if necessary. Do not round until the final calculation.
Provide your answer below:
5 of 5
(i) Let a discrete sample space be given by
Ω = {ω1, 2, 3, 4},
Total marks 12
and let a probability measure P on be given by
P(w1) 0.2, P(w2) = 0.2, P(w3) = 0.5, P(w4) = 0.1.
=
Consider the random variables X1, X2 → R defined by
X₁(w3) = 1, X₁(4) = 1,
X₁(w₁) = 1, X₁(w2) = 2,
X2(w1) = 2, X2(w2) = 2, X2(W3) = 1, X2(w4) = 2.
Find the joint distribution of X1, X2.
(ii)
[4 Marks]
Let Y, Z be random variables on a probability space (N, F, P).
Let the random vector (Y, Z) take on values in the set [0,1] × [0,2] and let the
joint distribution of Y, Z on [0,1] × [0,2] be given by
1
dPy,z(y, z)
(y²z + y²²) dy dz.
Find the distribution Py of the random variable Y.
[8 Marks]
Total marks 16
5.
Let (,,P) be a probability space and let X : → R be a random
variable whose probability density function is given by f(x) = }}|x|e¯|×| for
x Є R.
(i)
(ii)
Find the characteristic function of the random variable X.
[8 Marks]
Using the result of (i), calculate the first two moments of the
random variable X, i.e., E(X") for n = 1, 2.
(iii) What is the variance of X?
[6 Marks]
[2 Marks]
Chapter 3 Solutions
Probability and Statistical Inference (9th Edition)
Ch. 3.1 - Show that the mean, variance, and mgf of the...Ch. 3.1 - Let X be a continuous random variable with pdf...Ch. 3.1 - Customers arrive randomly at a bank tellers...Ch. 3.1 - If the mgf of X is M(t)=e5te4tt,t0andM(0)=1,, find...Ch. 3.1 - Let Y have a uniform distribution U(0,1), and let...Ch. 3.1 - A grocery store can order n watermelons from a...Ch. 3.1 - For each of the following functions, (i) find the...Ch. 3.1 - For each of the following functions, (i) find the...Ch. 3.1 - Prob. 9ECh. 3.1 - The pdf of X is f(x)=cx2,1x. (a) Calculate the...
Ch. 3.1 - The pdf of Y is g(y)=cy3,1y. (a) Calculate the...Ch. 3.1 - Sketch the graphs of the following pdfs and find...Ch. 3.1 - The logistic distribution is associated with the...Ch. 3.1 - Find the variances of each of the distributions...Ch. 3.1 - The life X (in years) of a voltage regulator of a...Ch. 3.1 - Let f(x)=(x+1)2,1x1. Find (a) 0.64, (b) q1=0.25,...Ch. 3.1 - An insurance agent receives a bonus if the loss...Ch. 3.1 - Prob. 18ECh. 3.1 - The total amount of medical claims (in $100000) of...Ch. 3.1 - Nicol (see References) lets the pdf of X be...Ch. 3.1 - Let X1,X2,...,Xk be random variables of the...Ch. 3.2 - Prob. 1ECh. 3.2 - Telephone calls arrive at a doctors office...Ch. 3.2 - Let X have an exponential distribution with mean...Ch. 3.2 - Let F(x) be the cdf of the continuous-type random...Ch. 3.2 - There are times when a shifted exponential model...Ch. 3.2 - A certain type of aluminum screen 2 feet in width...Ch. 3.2 - Find the moment-generating function for the gamma...Ch. 3.2 - If X has a gamma distribution with =4 and =2, find...Ch. 3.2 - If the moment-generating function of a random...Ch. 3.2 - Use the moment-generating function of a gamma...Ch. 3.2 - Let X have a gamma distribution with parameters ...Ch. 3.2 - Let X equal the number of alpha particle emissions...Ch. 3.2 - If X is x2(23), find the following: (a)...Ch. 3.2 - If X is x2(12), find constants a and b such that...Ch. 3.2 - Prob. 15ECh. 3.2 - Cars arrive at a tollbooth at a mean rate of five...Ch. 3.2 - If 15 observations are taken independently from a...Ch. 3.2 - Say the serum cholesterol level (X) of U.S. males...Ch. 3.2 - A bakery sells rolls in units of a dozen. The...Ch. 3.2 - The initial value of an appliance is $700 and its...Ch. 3.2 - A loss (in $100000) due to fire in a building has...Ch. 3.2 - Prob. 22ECh. 3.2 - Prob. 23ECh. 3.2 - Prob. 24ECh. 3.3 - If Z is N(0,1), find (a) P(0.47Z2.13). (b)...Ch. 3.3 - If Z is N(0,1), find (a) P(0Z0.78). (b) P(2.46Z0)....Ch. 3.3 - If Z is N(0,1), find values of c such that (a)...Ch. 3.3 - Find the values of (a) z0.10, (b) z0.05, (c)...Ch. 3.3 - If X is normally distributed with a mean of 6 and...Ch. 3.3 - If the moment-generating function of X is...Ch. 3.3 - If X is N(650,400), find (a) P(600X660). (b) A...Ch. 3.3 - Prob. 8ECh. 3.3 - Find the distribution of W=X2 when (a) X is...Ch. 3.3 - If X is N(,2) show that Y=aX+b is N(a,+b,a22),a0,...Ch. 3.3 - A candy maker produces mints that have a label...Ch. 3.3 - Prob. 12ECh. 3.3 - The serum zinc level X in micrograms per deciliter...Ch. 3.3 - The strength X of a certain material is such that...Ch. 3.3 - The fill problem is important in many industries,...Ch. 3.3 - The graphs of the moment-generating functions of...Ch. 3.3 - Prob. 17ECh. 3.4 - Let the life W (in years) of the usual family car...Ch. 3.4 - Suppose that the length W of a mans life does...Ch. 3.4 - Let Y1 be the smallest observation of three...Ch. 3.4 - Prob. 4ECh. 3.4 - Let X be a random variable of the mixed type...Ch. 3.4 - Let X be a random variable of the mixed type...Ch. 3.4 - Prob. 7ECh. 3.4 - Find the mean and variance of X if the cdf of X is...Ch. 3.4 - Consider the following game: A fair die is rolled....Ch. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Let X have an exponential distribution with =1;...Ch. 3.4 - A loss X on a car has a mixed distribution with...Ch. 3.4 - A customer buys a $1000 deductible policy on her...Ch. 3.4 - A certain machine has a life X that has an...Ch. 3.4 - Prob. 16ECh. 3.4 - Some banks now compound daily, but report only on...Ch. 3.4 - Prob. 18ECh. 3.4 - Prob. 19ECh. 3.4 - Let X be the failure time (in months) of a certain...Ch. 3.4 - In a medical experiment, a rat has been exposed to...
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