Consider two continuous random variables X and Y, where Y~U(1,3) and X|Y = y - U(0,1/y). (a) State E(X|Y = y) and hence, using the method of iterated expectations, find E(X) (b) Show that the joint probability density function of X and Y is fxx (x, y) = {'/2. P/2, 0
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- b) A random variable Y has a probability density function defined as: -y 1 f(y)= 0 = eLet (X,Y) be bivariate random variables having joint probability density function as Ex + y) 0sx< 2,0 sys2 f(x): otherwise Find the following: (i) The correlation coefficient between X and Y. (ii)E(X"Y*)Find the distribution of Z = X +Y, if X and Y are jointly continuous random variables with joint PDF given by fx.x(x,y) = 2e¬(s+v) I0.=|(14)[(0,20) (x). Let X and Y be two random variables with joint probability density function given by fxx(z,y) = 2e¬(=+v) 10,1(4)I[0,»)(1). Find the joint probability density function of X and X +Y. Let X1 and X2 be iid U(0,1) random variables. Find the joint probability density function of Y1 = X1 + X2 and Y2 = X2 – X1. %3D
- 4. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = gx (x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be - 1.²0⁰ with a similar definition for Y but g replaced by h and x replaced by y. We also define to be the expectation of XY. E[X] = xg(x) dx, E(XY)= (0,00)x(0,00) Tuf(x,y)dady (0,∞) (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y].The random variable Y has probability density function f(V) = k(y + y³), 0 2. Hence find PG < Y <). iii) Find the variance of Y.Let X,Y be two random variables with joint probability density function f(x.y) =0< xLet X and Y be continuous random variables having a joint probability density function (pdf) given by f(x, y) = e-y, (i) (ii) (iii) = 2|X = 3).Let X and Y be two continuous random variables with joint probability density [3x function given by: f(x,y)= 04. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = 9x(x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be E[X] = √rg(x)dx, to be the expectation of XY. 0 with a similar definition for Y but g replaced by h and x replaced by y. We also define E[XY] = (0,00)x (0,00) 110,00)x (0,00) 29 (x, y) dedy (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y]. [2]Let X and Y be independent normally distributed random variables with mean 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, where a + -1 is a real number. Find the absolute value of the Jacobian of transform from X, Y to U, V. (c) Find the joint probability density function for U and V. Find a for which U and V are independent random variables. Write down fu,v (u, v) for this a in the answer.Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.b) Let Y1, Y2.,Y5 be independent random variables with probability density function y 1 e 4 4 f(y)= ,y > 0 ,elsewhere 3 Determine the distribution and parameter of V = EY; using the method of moment i=1 generating function. Hence, find the mean of V using the moment generating function.SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,