A First Course In Probability, Global Edition
10th Edition
ISBN: 9781292269207
Author: Ross, Sheldon
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Question
Chapter 6, Problem 6.41P
To determine
To Show :The independence of X, Y & Z implies independence of X & Y.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
1. Let X₁ and X₂ be any random variables, then V(X₁ + X₂) must be:
A) greater than V(X₁) + V(X₂).
B) less than V(X₁) + V(X₂).
C) 0.
D) None of the above.
Prove the following
5. Let X1 - b(n1, p) and X2 ~ b(n2,p) be independent random variables.
a) Find the m.g.f. of Y = X1+ X2.
%3D
b) How is Y distributed?
Chapter 6 Solutions
A First Course In Probability, Global Edition
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
Similar questions
- Prove the followingarrow_forwardLet X and Y be independent discrete random variables and suppose that X+Y=2. Show that X and Y are constant random variables.arrow_forwardLet X1, X2, X3 be random variables such that Var(X1) = 5, Var(X2) = 4, Var(X3) = 7, cov(X1, X2) = 3, cov(X1, X3) = -2 and X2 and X3 are independent. Find the covariance between Y1 = X1 – 2X2 + 3X3 and Y2 = -2X1 + 3X2 + 4X3. %3Darrow_forward
- Let X1, X2, X3 be random variables such that V ar(X1) = 5, V ar(X2) = 4,V ar(X3) = 7, cov(X1, X2) = 3, cov(X1, X3) = −2 and X2 and X3 are independent. Findthe covariance between Y1 = X1 − 2X2 + 3X3 and Y2 = −2X1 + 3X2 + 4X3arrow_forwardA box contains 4 black balls and 6 white balls. A player randomly draws one ball and returns the ball to the box along with 2 additional balls with the same color. Then the player randomly draws another ball. Define the random variables as If the first ball is black, Y = If the first ball is white, If the second ball is black, Y, : If the second ball is white, a) Find E(Y,) and Var (Y,) b) Calculate P(Y,+Y, <1) c) Calculate covariance between Y and Y, 6 3 and 5 1 2 A) 5' 5 25 1 and 5 4 В) 25 5 25 1 and 25 5 25 4 1 C) 1 and 5 2 D) 25 5 25 2. 3.arrow_forwardii) Let X be a random variable taking three values: P(X= a₁) = P₁, P(X=a₂) = P2, P(X=a3) = P3, where p₁ + P2 + P3 = 1 and P₁, P2, P3 € (0, 1). Let A = {X = a₁} and G = {N, 0, A, Ac}. Prove that E (X³|G) = a³¹₁ + ª²P² + a²P3, P2 + P3 1A.arrow_forward
- Let X1, X2, and X3 be independent random variables such that E(X1) ‚E(X2) and E(X3) Further, suppose that Var(X1) = 5, Var(X2) = 10, and Var(X3) = 15. 1.) Compute E(TX1 + 4X2 + V3X3+ 15) 2.) Compute Var(/2X1 + V3X2 + 7X3+arrow_forwardWe have two independent random variables X, and X2. Let X,~N(5,15) and X2~x. Calculate P(X, - 5> 3, X2)-arrow_forward5) Let N= [0, 1) and let F = o ([0, 1/3), [1/3, 2/3), [2/3, 1)) (a) Write down F. (b) Is X(w) = 2w an F - random variable on N? Why? (c) Find all possible F - random variables on 2.arrow_forward
- Let Y₁ = (x,-X₂), where X, and X₂ are independent random variables and each of them X, 2 is distributed as x² (2). Find the p.d.f. of Y₁.arrow_forwardSuppose random variables X and Y are conditionally independent given random variable Z, i.e. P(X, Y|Z) = P(X|Z)P(Y|Z). Show that : (a) P(Y|X, Z) = P(Y|Z). (b) P(X|Y, Z) = P(X|Z).arrow_forwardThe CDF of a random variable X is given the function F(x)= cx² 3x² + x support of X is x = 1,2,3,... Determine P(X = 1) and P(X = 2). on the support of X, where thearrow_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