where , & 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.

1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.

ele) Consider the regression model Y; = BX; + u; Where u, and X, satisfy the assumptions specified here. Let p denote an estimator of p that is constructed as ß= where Y and X are the sample means of Y, and X, respectively. Show that B is a linear function of Y,, Y2, Y Show that B is conditionally unbiased. 1 E(Y,X,, X2.. X,) = ▼ 1 „P(X, +X2+ + X,) + Y2 + + Yn) (AX, + u) 2 E(BIX,. X2 X,) =E |(X, X2.. Xn) Click to select your answer(s). W DELL P Type here to search 1/5

Q. 3 Let continuous random variables X and Y be independent identically distributed random variables with the following respective pdfs: fx(x) = 2e-2x;x 2 0 and fy (y) = 2e-2y;y > 0 Define a new random variable Z = X + Y. Derive moment generating function of Z, i.e. Mz(t).