Let X and Y be two jointly continuous random variables with joint PDF y2 0 < x < 1,0 < y < 2 xy 4 4 fxx(x, y) : otherwise For 0 < y < 2, find a. the conditional PDF of X given Y = y; b. P(X < |Y = y).

Let X be a point randomly selected from the unit interval [0, 1]. Consider the random variable Y = (1-X)-¹/2 (a) Sketch Y as a function of X. (b) Find and plot the cdf of Y. (c) Derive the pdf of Y. (d) Compute the following probabilities: P(Y > 1), P(3 < Y < 6), P(Y ≤ 10).

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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).

If the joint pdf of the bivariate random variables X and Y is given by f(x,y) = 6 e-2x - 3Vx >0, y>0. Find the conditional pdf of Y given X Select one: а. f(y|x) =2 e-3y V20 b. f(y|x) = 3 e-2x x20 O C. f(y|x) =2 e-2x ,X20 O d. f(y|x) = 3e-3v V20

12. Let the random variable X and Y have joint pdf 4 f(x,y) = (x² + 3y²), 0