Let X and Y be independent normally distributed random variables withmean zero and variances og = 1 and of = 4.(a) Write the joint probability density function fx.y (r, y).• (b) Define new random variables U = aX + Y and V = X – Y, wherea + -1 is a real number. Find the absolute value of the Jacobian oftransform from X, Y to U, V.(c) Find the joint probability density function for U and V. Find a forwhich U and V are independent random variables. Write down fu,v (u, v)for this a in the answer.

Answered: Image /qna-images/answer/74222a4f-42a9-4c2f-9e17-ddd07a93f3b2.jpg

Answered: Let X and Y be independent normally…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.
b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.
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Suppose that X, Y are jointly continuous with joint probability density function f( x, y){ xe^-x(1+y), ifx >0 and y >00, otherwise. (a) Find the marginal density functions of X and Y. (b) Calculate the expectation E[XY]. (c) Calculate the expectation EIX/(1+ Y )1. (e) Determine if the random variables X and Y in this exercise are independent.
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1)Let x and y be two continuous random variables whose function is the probability density joint is given by : a)Draw the relationship between the variables x and y on the Cartesian axes .b)Calculate the marginal pdfs px(X)and py(Y).c)Are the v.a.s x and y independent ?
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