2. Let X and Y have the joint density given by: f(x, y) = k(2x+y) for 0 < x, y < 1 0 otherwise where k is a constant. Find k. Also, obtain the conditional density, conditional mean and conditional variance of Y given X = ½ (14 pointel

Don't hand writing solution.

4. Find the average value of the sum of the squares of two numbers x and y where (b) x + y| < 1 and x - y| < 1.

Let Ap and B be the random variables. A random process is defined a X () = Ag cos ont+ B sin ant, where og is a real constant. Find the power density spectru m of X (), if Ag and B, are uncorrelated random variables with zero mean and same variance.

Find the density of U = Y1 +Y2, where Y1 and Y2 are independent random variables with densities (yı – 1), 3 < Yı < 4, (y2 + 2), 0 < Y2 < 1, fy, (y1) : fy,(y2) = otherwise, otherwise.

Let X and Y be independent and normally distributed RVs with the same PDF Find the PDFS of: (i) (ii) x?: (iii) x²+ Y?. X + Y;

Assume that x,, . .. , Xn are the realization of the random samples X, ... , Xn with pdf or pmf fo. Find the UMP of Ho : 0 = 0 versus H, : 0 = 0, . for 0, > 0o fe(x) is density of normal with mean 0 and known o. fe(x) is density of normal with known mean u and variance 0 = o² .