a) Find the conditional density of X given Y = y. b) Find E(ex|Y).
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- 1)A random vector z = (x y) T has joint probability density given by:a)Sketch the graph of the probability density function pxy(X, Y ).b)Determine the value of the constant C.c)Determine the mean mz vector.Exercise 40 Let X Unif(0,1). Let g(x) = e" and Y = g(X). (i) Find the density fx of the random variable X. (ii) Find the cdf Fy (y) = P(Y < y) of Y. (iii) Find the density fy of the random variable Y.c) Let X [0,1]x[0,1]x[0,1]. Let Y = g(X), be a two dimensional random vector, where 91(X ) = X1 + X2 , g2(X) = X2 – X3 . Find the density function of Y. [X1,X2, X3] be a three dimensional random vector uniformly distributed on
- 3. Let f be the density function of a random variable X, and let Y = (Y1, Y2, Y3) be a continuous random vector with density function %3! fY (y1, V2, ys) 6 f(v)f(v)/() 1{vn < ya < ya}. (a) Find the conditional density fy, Y,Y, (V, Vsly2) and argue that (Y I. Y) | Y. (b) If X has a uniform distribution on the unit interval (0, 1], deduce that (Y, Y) | (Y = V) has a certain bivariate uniform distribution. Over which set?3. Let random variables X and Y be independent with joint density fe(x, y). Let Ix (0) and Iy (0) be the Fisher information of X and Y, respectively. Prove that the Fisher information I(x,y)(0) = Ix(0) + Iy(0).Let the r.v X have the pdf f(x) = 4x°, 0If the joint probability density of X and Y is given by:f(x, y) = 6y if 0 < x < 1, 0 < y < 1, x + y < 1 0 otherwise. a)Find marginal densities of X and Y b) Are X and Y are independent? c)Find the conditional probability of Y given that X = 1/2. d) Find covariance of r.v.’s X and Y , i.e., cov(X, Y ). Comment on their correlation based on covariancevalue.1) Consider a continuous RV, Y, with a pdf given by f(y)=ce²¹, -∞(s) Find the linearization of f at the given point. Then estimate the value of f. (a) f(x, y) = xy² at (2, 1). Estimate f(2.001, 1.05) y? f(x, y) = at (1,2). Estimate f(0.99, 1.99)Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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