8. A pair of jointly continuous random variables, X and Y have the joint density 10 function uniform distributed over 1/2
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![8. A pair of jointly continuous random variables, X and Y have the joint density 10
function uniform distributed over 1/2<x+y<1, if 0 < x, y ≤1 and 0 elsewhere.
(a) Find the PDF, fxy(x, y).
(b) Find the marginal PDFs of X and Y, i.e. fx (x) and fy (y).
(c) Find E[XY ] and Var[X]
(d) Find the conditional PDF for X given that Y3/4, i.e., Jxy (2|3/4).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcd2d1959-46ea-4e22-9d0f-1ad88e6676e4%2Fc579ac10-448a-479e-b7e9-4216a88282c7%2Fb61mtf_processed.png&w=3840&q=75)
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- (i) Continuous RVs X and Y have a joint PDF (m+n+2)! S(x, y) = (1-x)"y". m!n! for 0 < y2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 04. Let X and Y be independent, continuous random variables with densities and fx(x) = = fy (y) = = 0 {} if 0 < x < 2 otherwise. y if 0Let X and Y random variables have independent Gamma distributions with X-Gamma(1, 6) and Y-Gamma(2, B). a. Find the joint probability density of Z, = X + Y, Z, = X+Y a. Find the marginal pdf of Z2.We have a random variable X and Y that jave the joint pdf f(x) = {1 0<x<1, 0<y<1} {0 otherwise} Let U = Y - X2. What is the support for the random variable U? Are there critical points? If U = Y/X. What is the support for the random variable U? Are there critical points?27. If the joint density of the random variables X and Y is [emin{x,y) - 1] e-(x+y) if 0 < x, y <∞ f(x, y) = 0 otherwise, then what is the marginal density function of X, where nonzero?Let (X, Y) be a pair of jointly continuous random variables with joint p.d.f. 3 f (x, y) : =, 0sx< 1, 2²< y< 1. 2' (a) Find the marginal p.d.f. fx(x) of X and the marginal p.d.f. fy (y) of Y. (Ь) Find the covariance Cov(X, Y) of X and Y. (c) Find the conditional p.d.f. fy|x(y|x) of Y given X = x, for any x E (0, 1). (d) Find the conditional expectation E(Y|X = 1/2) of Y given X = 1/2. 2.The probability density function of a distribution is given by x f(x) = exp(-7). Use differentiation to show that the probability density function has a maximum at x = 0. The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1], q = 1 - p and n is a positive integer. Use the moment generating function to find the mean and variance of the distribution in terms of p, q and n.4) The joint density function of the random variables X and Y is given by (&xy f(x, y) = 0SXS1,0 sysx otherwise Find (a) the marginal density of X, (b) the marginal density of Y, (c) the conditional density of X, (d) the conditional density of Y.1. The joint density for the random variables (X, Y) is given below. for 0SS1,0SySI 0. for clsewhere (a) Find the marginal density of X (b) Find the marginal density of Y (c) Find VarlX] (d) Find VarV] (e) Are X and Y statistically independent? () Find P(X>0.6) (R) I EXY-0.8, find the correlation between X and Y12. Let the random variable X and Y have joint pdf 4 f(x,y) = (x² + 3y²), 0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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