Problem 8 (p.204#15). Show the following.a) If X and Y are independent random variables,V(X-Y) = V(X) +V(Y)b) Let D₁ and D₂ represent two draws at random with replacement from a population withE(D₁) = 10, S(D₁) = 2. Find a number c such thatP(|D₁ - D₂|

Given: (a) X and Y are independent random variables. Required: To prove VX-Y=VX+VY

Answered: Problem 8 (p.204#15). Show the…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

3. Let X and Y be two independent Bernoulli (0.5) random variables. Let Z = X + Y. Show that Var(Z) = E[Var (ZIX)] +Var(E[ZIX]).
2. If X is a random variable taking values 0,1,2,.... and P(X) = ab*, where a and b are positive constants such that a+b= 1, then (i) Find mgf. (ii) If E(X) = m₁, E(X2) = m2, show that m₂ = m₁(2m₁ + 1).
Do #3
1. Let X be a random variable having pdf f(x) = 6x(1 – x) for 0 < I < 1 and 0 elsewhere. Compute the mean and variance of X. 2. Let X1, X2,..., X, be independent random variables having the same distribution as the variable from problem 1, and let X, = (X1+ ·.+Xn). Part a: Compute the mean and variance of X, (your answer will depend on n). Part b: If I didn't assume the variables were independent, would the calculation in part a still work? Or would at least part of it still work? 3. Suppose that X and Y are both independent variables, and that each has mean 2 and variance 3. Compute the mean and variance of XY (for the variance, you may want to start by computing E(X²Y²)). 4. Suppose that (X,Y) is a point which is equally likely to be any of {(0, 1), (3,0), (6, 1), (3, 2)} (meaning, for example, that P(X = 0 and Y = 1) = }). Part a: Show that E(XY) = E(X)E(Y). Part b: Are X and Y independent? Explain. 5. Let X be a random variable having a pdf given by S(2) = 2e-2" for 0
1) Assume a relationship Y = g2(91(X)) between the two random variables X and Y where gi and g2 are two functions. To compute the mean of Y, we can choose any of the following equivalent expressions: E(Y)= | fr (u)ydy = | fn(x)(u)g2(u)du = / fx(#)g2(9n(x)dr (1) -00 where fy (y), f91(x)(u) and fx(x) are the probability density functions (p.d.f.s) of Y, 91(X) and X, respectively. Now consider Y = X? where X is N(0, o2). a) Determine the p.d.f. fy (y) of Y. Hint: for y > 0, fy (y) Prob(Y < y) = Prob(-< X < VI) = % x (x)dx. b) Then evaluate explicitly E(Y)= S fy(y)ydy.
2. (Corrected.) Suppose Y is a random variable with P{Y = -2} = 2/10, P{Y=2} = 5/10 and P{Y = 3} = 3/10. (a) Compute the mean of Y. (b) Compute E[Y²]. (c) Compute the variance of Y. (d) Compute E[Y+]. (e) Compute E[|Y|]. (f) Compute E[Y^2].
Suppose that X is a random variable for which the m.g.f. is as follows: M(t) = 1/4(3et+e-t), -∞<t<∞ Find the mean and variance of X.
Suppose we have three independent random variables X, Y and Z where... Var(X + 2Y) = 13, Var(2Y + 4Z) = 40, and Var(Z) = 2. a) If E(Y) = E(Z) = 1, what is E(Y2+Z2)? b) What is SD(X + Y + Z)?
2. Let X be a random variable with p.m.f., (1+ x? f(x) = -,x = -1,0,1,2,3 20 0 ,otherwise. Find E(3X2 + 6).
4. Suppose X₁ and X₂ are independent random variables with cdf Fx(x) = sin(x), 0 ≤ x ≤ 1/ a. Show that fx(2) (x) = 2sin(x) cos(x), where X (2) is the random variable for the 2nd order statistic. (Remember, we only have 2 random variables.) b. Show that fx(1)(x) = 2cos(r) [1 sin(x)], where X(1) is the random variable for the 1st order statistic.
1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).
7. A discrete random variable X taking non-negative integer values has probability gen- erating function: 1 rx(t) = (2- t)2 (a) Calculate P(X = 2).

SEE MORE QUESTIONS

Recommended textbooks for you

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

SEE MORE TEXTBOOKS

Problem 8 (p.204#15). Show the following. a) If X and Y are independent random variables, V(X-Y)= V(X) +V(Y) b) Let D₁ and D₂ represent two draws at random with replacement from a population with E(D₁)= 10, S(D₁) = 2. Find a number c such that P(|D₁ - D₂|