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- Suppose that Y is a continuous random variable. Show EY yfr(y)dy.Let Xand Y be two continuous random variables with joint probability density [3x function given by: f(x.y)%D 0sysxsl elsewhere with E(X) = ECX)- EC) - EC*)= ;and E(XY) = 10 3 E(Y*) = - and E(XY) =; %3D Then the value of the variance of 2X+Y is: O 3/80 O 91/320 43/320 7/20Exercise 20. 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.
- Let X be a (continuous) uniform random variable on the interval [0,1] and Y be an exponential random variable with parameter lambda. Let X and Y be independent. What is the PDF of Z = X + Y.2. If three random variables have the joint density S{(r + y)e-, for 0 0 10, f(x, y, 2) = elsewhere (a) find the probability that X 1; (b) check whether the three random variables are independent%3; (c) check whether any two of the three random variables are pairwise independent.Suppose that X and Y have a joint probability density function given by ce-3z-5y if r, y 20 fx.x(T, y) = otherwise Are the random variables X and Y statistically independent? Justify your answer.
- Answer all three questions in this section 2. Let V and W be two random variables with joint probability density function given by fvw (v,w) = aw exp(-dv +(8-A)w²), w>0, v> w², where A, & are positive constants. (a) Find the value of a. (b) Consider the transformation X = WP, Y=V_W². Find the joint density of X and Y. Hence show that X and Y are independent exponential random variables. (c) State the distribution of AX + SY.3. Consider the joint density of two random variables X, Y, f(x, y) = { 6(1 – y) if 0Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.4) The joint probability density function of two random variables is: fxy(x, y) = {c(1 + xy) 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2 elsewhere a) Find Fxy (0.5, 1.0). b) Find fxy(x, 1). c) Find fxy(x | 1).2) The joint probability density function of random variables X and Y is Sx,(x, y) ={ fxy 0sxs1, 0sy<2, 0 otherwise. Find the following quantities. a) E[X] and o;. b) E[Y] and o;. c) The correlation Ry = E[XY]. d) The covariance Cxy. e) The correlation coefficient Px.x:X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2SEE MORE QUESTIONS