EBK FIRST COURSE IN PROBABILITY, A
10th Edition
ISBN: 9780134753676
Author: Ross
Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)
expand_more
expand_more
format_list_bulleted
Question
Chapter 8, Problem 8.6TE
a.
To determine
To show:
b.
To determine
To show:
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Let X and Y be continuous random variables with joint distribution function, F (x,y).
Let g (X,Y) and h (X,Y) be functions of X and Y.
PROVE
Cov (X,Y) = E[XY] - E[X] E[Y]
7.
Let X and Y denote two continuous random variables. Let f(x,y) denote the
joint probability density function and fx(x) and fy (y) the marginal probability
density functions for X and Y, respectively. Finally let Z = aX + bY, where a
and b are non-zero real numbers.
(e)
Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y).
[You may use standard results relating to variance and covariance without
proof, but these should be clearly stated.]
Let X be a random variable on a closed and bounded interval [a, b]. Let g(x) be a convex function. Prove that
g(E(X)) ≤ E (g(X)
Chapter 8 Solutions
EBK FIRST COURSE IN PROBABILITY, A
Ch. 8 - Suppose that X is a random variable with mean and...Ch. 8 - From past experience, a professor knows that the...Ch. 8 - Use the central limit theorem to solve part (c) of...Ch. 8 - Let X1,...,X20 be independent Poisson random...Ch. 8 - Fifty numbers are rounded off to the nearest...Ch. 8 - A die is continually rolled until the total sum of...Ch. 8 - A person has 100 light bulbs whose lifetimes are...Ch. 8 - In Problem 8.7, suppose that it takes a random...Ch. 8 - If X is a gamma random variable with parameters...Ch. 8 - Civil engineers believe that W, the amount of...
Ch. 8 - Many people believe that the daily change of price...Ch. 8 - We have 100 components that we will put in use in...Ch. 8 - Student scores on exams given by a certain...Ch. 8 - A certain component is critical to the operation...Ch. 8 - An insurance company has 10.000 automobile...Ch. 8 - A.J. has 20 jobs that she must do in sequence,...Ch. 8 - Redo Example 5b under the assumption that the...Ch. 8 - Repeat part (a) of Problem 8.2 when it is known...Ch. 8 - A lake contains 4 distinct types of fish. Suppose...Ch. 8 - If X is a nonne9ative random variable with mean...Ch. 8 - Let X be a nonnegative random variable. Prove that...Ch. 8 - Prob. 8.22PCh. 8 - Let X be a Poisson random variable with mean 20....Ch. 8 - Prob. 8.24PCh. 8 - Prob. 8.25PCh. 8 - If f(x) is an Increasing and g(x) is a decreasing...Ch. 8 - If L(p) is the Lorenz curve associated with the...Ch. 8 - Suppose that L(p) is the Lorenz curve associated...Ch. 8 - If X has variance 2, then , the positive square...Ch. 8 - If X has mean and standard deviation , the ratio...Ch. 8 - Compute the measurement signal-to-noise ratio-that...Ch. 8 - Let Zn,n1, be a sequence of random variables and...Ch. 8 - Prob. 8.5TECh. 8 - Prob. 8.6TECh. 8 - Prob. 8.7TECh. 8 - Explain why a gamma random variable with...Ch. 8 - Prob. 8.9TECh. 8 - If X is a Poisson random variable with mean , show...Ch. 8 - Prob. 8.11TECh. 8 - Prob. 8.12TECh. 8 - Prob. 8.13TECh. 8 - Prob. 8.14TECh. 8 - If f and g are density functions that are positive...Ch. 8 - Prob. 8.16TECh. 8 - The number of automobiles sold weekly at a certain...Ch. 8 - Prob. 8.2STPECh. 8 - If E[X]=75E[Y]=75Var(X)=10var(Y)=12cov(X,Y)=3 give...Ch. 8 - Prob. 8.4STPECh. 8 - Prob. 8.5STPECh. 8 - Prob. 8.6STPECh. 8 - Prob. 8.7STPECh. 8 - Prob. 8.8STPECh. 8 - Prob. 8.9STPECh. 8 - A tobacco company claims that the amount of...Ch. 8 - Prob. 8.11STPECh. 8 - Prob. 8.12STPECh. 8 - The strong law of large numbers states that with...Ch. 8 - Each new book donated to a library must be...Ch. 8 - Prove Chebyshevs sum inequality, which says that...
Knowledge Booster
Similar questions
- Let X be a continuous random variable with cdf F(x). Show that E(I(X < x)) = F(r) where I is the indicator function (1 if Xarrow_forwardPlz do fastarrow_forwards) Let X be a discrete random variable taking values {x1, x2, . . . , xn} with probability {p1, p2, . . . , pn}. The entropyof the random variable is defined asH(X) = − Σn i=1 pi*log(pi)Find the probability mass function for the above discrete random variable that maximizes the entropy.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_forwardb) Suppose that a sequence of mutually independent and identically distributed continuous random variables X₁, X₂, X3,..., X has the probability density function -√√2+² i) 2 foro < x < 0⁰ elsewhere Show that f(x; 6) belongs to the one-parameter regular exponential family. Clearly indicate the following functions, h(x), c(0), w(0) and t(x). ii) Show that ΣX, is the maximum likelihood estimator of the parameter 0. iii) is the maximum likelihood estimator 6=EX₁, an efficient estimator of 6? _(x-0)² f(x; 0)=√√√2arrow_forward3. Let X, Y be independent, standard normal random variables. (a) What is the probability density function of aX + bY? (b) Compute E[X|X – 2Y] and E[X²|X – 2Y].arrow_forwarda) Let X be a continuous random variable with the following probability density function (2x3 f(x)={11 2 +5) , 0arrow_forwarda) Suppose that, for -1 < a < 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {11-a(1- ([1-a(1-2e-x¹)(1-2e-*²)]e*¹*₂ otherwise ,0 ≤ x1,0 ≤ x₂. i) Find the marginal distribution of X₁. ii) Find E(X₂X₂).arrow_forwardLet X and Y be independent uniform random variables on (0,1). Let Z = [1/(X+Y)]. (Recall that is the largest integer ≤ x.) (a) Find P(Z = 0). (b) Find E[Z].arrow_forwardLet XE {-1,0, 1} . That is, X is a discrete random variable only takes three values -1, 0, and 1. Suppose the equality for Chebyshev's inequality holds for X and P(X = 0) = 0.3 , find P(X = 1) Let X be a random variable with probability density f(x) = x6 for x > 1 and O else. Use Chebyshev's inequality to bound P(X > 2.5) . Round your answer to 3 decimal places.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) = gx (x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be - 1.²0⁰ with a similar definition for Y but g replaced by h and x replaced by y. We also define to be the expectation of XY. E[X] = xg(x) dx, E(XY)= (0,00)x(0,00) Tuf(x,y)dady (0,∞) (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].arrow_forwardX1 and X2 are independent random variables with X1 ~ Exp(A1) and X2 ~ Exp(A2). Let M = min{X1, X2}. Determine the probability density function of M.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON