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Consider the random variable X whose probability density function is given by k/x3 if x>r fx(x) = otherwise Suppose that r=5.2. Find the value of k that makes fx(x) a valid probability density function.
3. Let X has the following pdf: {. -1 <1 fx(a) otherwise 1. Find the pdf of U X2. 2. Find the pdf of W X
Let X, Y be jointly continuous with joint density function (pdf) fx,y(x, y) *(1+xy) 05 x <1,0 <2 0 otherwise (a) Find the marginal density functions (pdf) fx and fy. (b) Are X and Y independent? Why or why not?
(5 pts) Consider a random variable X with pdf V o S 0.5eX fx(x) = { 0.5e-2 x < 0 x > 0. 0.5 e-> o ΔΙ Let S 4x2 x < 0 g(x) = { 22 x 20, V ΔΙ and let Y = g(X). Determine fy(y).
3. Suppose that X has pdf fx(x) = 3, x > 1 and Y has pdf 24» fy(y) = ¡2, x 〉 1. Suppose further that X and Y are inde- pendent. Calculate the P(X 〈 Y).
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
The PDF of random variable X and the conditionalPDF of random variable Y given X are fX(x) = 3x2 0≤ x ≤1, 0 otherwise, fY|X(y|x) = 2y/x2 0≤ y ≤ x,0 < x ≤ 1, 0 otherwise. (1) What is the probability model for X and Y? Find fX,Y (x, y). (2) If X = 1/2, nd the conditional PDF fY|X(y|1/2). (3) If Y = 1/2, what is the conditional PDF fX|Y (x|1/2)? (4) If Y = 1/2, what is...
Let X and Y be a random variable with joint PDF: { ay fxy (x, y) x > 1,0 <y <1 0 otherwise x2, 1. What is a? 2. What is the conditional PDF fy\x(x|y) of Y given X = x? 3. What is the conditional expectation of Ygiven X? 4. What is the expected value of Y?
4.5.4 X and Y are random variables with the joint PDF ( 5x2/2 JX,Y (x, y) = -1 < x < 1; 0 <y < x2, otherwise. 10 (a) What is the marginal PDF fx(x)? (6) What is the marginal PDF fy(y)?
7. The random variables X and Y have joint probability density function f given by 1 for x > 0, |y| 0 otherwise. 1-x, Below you find a diagram highlighting the (r, y) pairs for which the pdf is 1 (a) Calculate the marginal probability density function fx of X (b) Calculate the marginal cumulative distribution function Fy of Y (c) Are X and Y independent? Explain.