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, 01) Prove that the density for the standard normal distribution Satisfies: f(x) = 1 √2π ∞ 1₁ -8 f(x) dx = 1Show V(Y)The continuous random variables X and Y are statistically independent and have marginal density functions fx(x) = 2x, 0 1. y2' Calculate the probability P(X < 0.5, Y < 2)Consider two continuous random variables X and Y with marginal distributions g(x) and h(y)respectively and the joint density function given by: x > 0, 0 < y < 2 elsewhere. f (r.y) Then: X and Y ar statistically dependent f(x,y)#g(x)h(y) None of these f(ylx)=g(x)Suppose 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 027. 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. 2.4) 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 YRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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