FIRST COURSE IN PROBABILITY (LOOSELEAF)
10th Edition
ISBN: 9780134753751
Author: Ross
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 6, Problem 6.29TE
Show that the
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
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 6 Solutions
FIRST COURSE IN PROBABILITY (LOOSELEAF)
Ch. 6 - Two fair dice are rolled. Find the joint...Ch. 6 - Suppose that 3 balls are chosen without...Ch. 6 - In Problem 8 t, suppose that the white balls are...Ch. 6 - Repeat Problem 6.2 when the ball selected is...Ch. 6 - Repeat Problem 6.3a when the ball selected is...Ch. 6 - The severity of a certain cancer is designated by...Ch. 6 - Consider a sequence of independent Bernoulli...Ch. 6 - Prob. 6.8PCh. 6 - The joint probability density function of X and Y...Ch. 6 - Prob. 6.10P
Ch. 6 - In Example Id, verify that f(x,y)=2exe2y,0x,0y, is...Ch. 6 - The number of people who enter a drugstore in a...Ch. 6 - A man and a woman agree to meet at a certain...Ch. 6 - An ambulance travels back and forth at a constant...Ch. 6 - The random vector (X,Y) is said to be uniformly...Ch. 6 - Suppose that n points are independently chosen at...Ch. 6 - Prob. 6.17PCh. 6 - Let X1 and X2 be independent binomial random...Ch. 6 - Show that f(x,y)=1x, 0yx1 is a joint density...Ch. 6 - Prob. 6.20PCh. 6 - Let f(x,y)=24xy0x1,0y1,0x+y1 and let it equal 0...Ch. 6 - The joint density function of X and Y is...Ch. 6 - Prob. 6.23PCh. 6 - Consider independent trials, each of which results...Ch. 6 - Suppose that 106 people arrive at a service...Ch. 6 - Prob. 6.26PCh. 6 - Prob. 6.27PCh. 6 - The time that it takes to service a car is an...Ch. 6 - The gross daily sales at a certain restaurant are...Ch. 6 - Jills bowling scores are approximately normally...Ch. 6 - According to the U.S. National Center for Health...Ch. 6 - Monthly sales are independent normal random...Ch. 6 - Let X1 and X2 be independent normal random...Ch. 6 - Prob. 6.34PCh. 6 - Teams 1, 2, 3, 4 are all scheduled to play each of...Ch. 6 - Let X1,...,X10 be independent with the same...Ch. 6 - The expected number of typographical errors on a...Ch. 6 - The monthly worldwide average number of airplane...Ch. 6 - In Problem 6.4, calculate the conditional...Ch. 6 - In Problem 6.3 calculate the conditional...Ch. 6 - Prob. 6.41PCh. 6 - Prob. 6.42PCh. 6 - Prob. 6.43PCh. 6 - The joint probability mass function of X and Y is...Ch. 6 - Prob. 6.45PCh. 6 - Prob. 6.46PCh. 6 - An insurance company supposes that each person has...Ch. 6 - If X1,X2,X3 are independent random variables that...Ch. 6 - Prob. 6.49PCh. 6 - If 3 trucks break down at points randomly...Ch. 6 - Consider a sample of size 5 from a uniform...Ch. 6 - Prob. 6.52PCh. 6 - Let X(1),X(2),...,X(n) be the order statistics of...Ch. 6 - Let Z1 and Z2 be independent standard normal...Ch. 6 - Derive the distribution of the range of a sample...Ch. 6 - Let X and Y denote the coordinates of a point...Ch. 6 - Prob. 6.57PCh. 6 - Prob. 6.58PCh. 6 - Prob. 6.59PCh. 6 - Prob. 6.60PCh. 6 - Repeat Problem 6.60 when X and Y are independent...Ch. 6 - Prob. 6.62PCh. 6 - Prob. 6.63PCh. 6 - In Example 8b, let Yk+1=n+1i=1kYi. Show that...Ch. 6 - Consider an urn containing n balls numbered 1.. .....Ch. 6 - Suppose X,Y have a joint distribution function...Ch. 6 - Prob. 6.2TECh. 6 - Prob. 6.3TECh. 6 - Solve Buffons needle problem when LD.Ch. 6 - If X and Y are independent continuous positive...Ch. 6 - Prob. 6.6TECh. 6 - Prob. 6.7TECh. 6 - Let X and Y be independent continuous random...Ch. 6 - Let X1,...,Xn be independent exponential random...Ch. 6 - The lifetimes of batteries are independent...Ch. 6 - Prob. 6.11TECh. 6 - Show that the jointly continuous (discrete) random...Ch. 6 - In Example 5e t, we computed the conditional...Ch. 6 - Suppose that X and Y are independent geometric...Ch. 6 - Consider a sequence of independent trials, with...Ch. 6 - If X and Y are independent binomial random...Ch. 6 - Suppose that Xi,i=1,2,3 are independent Poisson...Ch. 6 - Prob. 6.18TECh. 6 - Let X1,X2,X3 be independent and identically...Ch. 6 - Prob. 6.20TECh. 6 - Suppose that W, the amount of moisture in the air...Ch. 6 - Let W be a gamma random variable with parameters...Ch. 6 - A rectangular array of mn numbers arranged in n...Ch. 6 - If X is exponential with rate , find...Ch. 6 - Suppose thatF(x) is a cumulative distribution...Ch. 6 - Show that if n people are distributed at random...Ch. 6 - Suppose that X1,...,Xn are independent exponential...Ch. 6 - Establish Equation (6.2) by differentiating...Ch. 6 - Show that the median of a sample of size 2n+1 from...Ch. 6 - Prob. 6.30TECh. 6 - Compute the density of the range of a sample of...Ch. 6 - Let X(1)X(2)...X(n) be the ordered values of n...Ch. 6 - Let X1,...,Xn be a set of independent and...Ch. 6 - Let X1,....Xn, be independent and identically...Ch. 6 - Prob. 6.35TECh. 6 - Prob. 6.36TECh. 6 - Suppose that (X,Y) has a bivariate normal...Ch. 6 - Suppose that X has a beta distribution with...Ch. 6 - 6.39. Consider an experiment with n possible...Ch. 6 - Prob. 6.40TECh. 6 - Prob. 6.41TECh. 6 - Each throw of an unfair die lands on each of the...Ch. 6 - The joint probability mass function of the random...Ch. 6 - Prob. 6.3STPECh. 6 - Let r=r1+...+rk, where all ri are positive...Ch. 6 - Suppose that X, Y, and Z are independent random...Ch. 6 - Let X and Y be continuous random variables with...Ch. 6 - The joint density function of X and Y...Ch. 6 - Consider two components and three types of shocks....Ch. 6 - Consider a directory of classified advertisements...Ch. 6 - The random parts of the algorithm in Self-Test...Ch. 6 - Prob. 6.11STPECh. 6 - The accompanying dartboard is a square whose sides...Ch. 6 - A model proposed for NBA basketball supposes that...Ch. 6 - Let N be a geometric random variable with...Ch. 6 - Prob. 6.15STPECh. 6 - You and three other people are to place bids for...Ch. 6 - Find the probability that X1,X2,...,Xn is a...Ch. 6 - 6.18. Let 4VH and Y, be independent random...Ch. 6 - Let Z1,Z2.....Zn be independent standard normal...Ch. 6 - Let X1,X2,... be a sequence of independent and...Ch. 6 - Prove the identity P{Xs,Yt}=P{Xs}+P{Yt}+P{Xs,Yt}1...Ch. 6 - In Example 1c, find P(Xr=i,Ys=j) when ji.Ch. 6 - A Pareto random variable X with parameters a0,0...Ch. 6 - Prob. 6.24STPECh. 6 - Prob. 6.25STPECh. 6 - Let X1,...,Xn, be independent nonnegative integer...
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.Similar questions
- Total marks 16 5. Let (N,F,P) be a probability space and let X : N → R be a random variable such that the probability density function is given by f(x)=ex for x € R. (i) Find the characteristic function of the random variable X. [8 Marks] (ii) 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]arrow_forward6. Let P be the standard normal distribution, i.e., P is the proba- bility measure on (R, B(R)) given by 1 dP(x) = 를 = e dx. √2πT Consider the random variables 21 fn(x) = (1 + x²) en+2, x Є R, n Є N. Using the dominated convergence theorem, prove that the limit Total marks 9 exists and find it. lim E(fn) n∞ [9 Marks]arrow_forward6. Let X be a random variable taking values in (0,∞) with proba- bility density function fx(u) = 5e5u u > 0. Total marks 8 Let Y = X2. Find the probability density function of Y. [8 Marks]arrow_forward
- 5. Let a probability measure P on ([0,3], B([0,3])) be given by 1 dP(s): = ½ s² ds. 9 Consider a random variable X : [0,3] → R given by X(s) = s², sc [0,3]. S Total marks 7 Find the distribution of X. [7 Marks]arrow_forwardProblem: The probability density function of a random variable is given by the exponential distribution Find the probability that f(x) = {0.55e−0.55x 0 < x, O elsewhere} a. the time to observe a particle is more than 200 microseconds. b. the time to observe a particle is less than 10 microseconds.arrow_forwardProblem: The probability density function of a random variable is given by the exponential distribution Find the probability that f(x) = {0.55e-0.55 x 0 < x, O elsewhere} a. the time to observe a particle is more than 200 microseconds. b. the time to observe a particle is less than 10 microseconds.arrow_forward
- Unknown to a medical researcher, 7 out of 24 patients have a heart problem that will result in death if they receive the test drug. 5 patients are randomly selected to receive the drug and the rest receive a placebo. What is the probability that less than 4 patients will die? Express as a fraction or a decimal number rounded to four decimal places.arrow_forwardWas wanting to check if my calculations were correct Suppose 52% of the population has a college degree. If a random sample of size 808 is selected, what is the probability that the proportion of persons with a college degree will be less than 54%? Round to four decimal places. after following the formula I got 0.8724arrow_forwardAt the beginning of each semester, students at the University of Minnesota receive one prepaid copy card that allows them to print from the copiers and printers on campus. The amount of money remaining on the card can be modeled by a linear equation where A represents how much remains on the card (in dollars) and p represents the number of pages that the student has printed. The graph of this linear equation is given below. 100 90 80 70 60 50 40 30 20 10 0 A = Amount on Card ($) 0 200 400 600 800 1000 1200 1400 1600 p = Number of Pages Printed What information does the vertical intercept tell you (represent) for this problem? Be sure to include specific details in your answer -- your answer should have both quantitative and qualitative data to describe the answer in terms of the question.arrow_forward
- Data management no 2 thanksarrow_forwardG12 Data Management please help on the first question no 1 belowarrow_forwardTotal marks 14 4. Let X and Y be random variables on a probability space (N, F, P) that take values in [0, ∞). Assume that the joint density function of X and Y on [0, ∞) × [0, ∞) is given by f(x, y) = 2e-2x-y Find the probability P(0 ≤ X ≤ 1,0 ≤ y ≤ 2). (ii) spectively. [6 Marks] Find the the probability density function of X and Y, re- [5 Marks] 111) Are the X and Y independent? Justify your answer! [3 Marks]arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:9780134753119
Author:Sheldon Ross
Publisher:PEARSON
Statistics 4.1 Point Estimators; Author: Dr. Jack L. Jackson II;https://www.youtube.com/watch?v=2MrI0J8XCEE;License: Standard YouTube License, CC-BY
Statistics 101: Point Estimators; Author: Brandon Foltz;https://www.youtube.com/watch?v=4v41z3HwLaM;License: Standard YouTube License, CC-BY
Central limit theorem; Author: 365 Data Science;https://www.youtube.com/watch?v=b5xQmk9veZ4;License: Standard YouTube License, CC-BY
Point Estimate Definition & Example; Author: Prof. Essa;https://www.youtube.com/watch?v=OTVwtvQmSn0;License: Standard Youtube License
Point Estimation; Author: Vamsidhar Ambatipudi;https://www.youtube.com/watch?v=flqhlM2bZWc;License: Standard Youtube License