51. Calculate P(0 < x < 2; 1 < Y < 2), where X and Y have joint prob- ability density function (2xy + 2x + y) if0 < x < 4 and (0 < y < 2 p(x, y) = otherwise
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- Suppose X, Y have joint density function ƒ (x, y) = { # (ªy+y²), 0≤x≤1 and 0 ≤ y ≤1 0, (a) Check that fis a genuine joint density function. (b) Find the marginal density functions of X and Y (c) Calculate the probability P(X6. The joint probability density function of X and Y is given by 6 f(z,v) = (- ÷(² +).0 Y). (c) Find P(Y > |X < }). (d) Find E[X].6. Let f(x1,x2) = 1²e¬Mx1+*2), Let Y1 = X1 + X2 and X2 = Y2. (a) Find the joint distribution of Y1 and Y2: f(y1, y2) (b) Then compute the marginal density of Y1Q2. Here, the joint density function of two random variable X and Y are, S(2,9) = {clE*y* 3y² + x²y³), 1Calculate the probability that X + Y < 3, where X and Y have joint prob- ability density (2xy +2x + y) 0 < x < 3, 0 < y < 3 p(x, y) = otherwise(i) Continuous RVs X and Y have a joint PDF (m+n+2)! S(x, y) = (1-x)"y". m!n! for 0 < ySuppose that an automobile dealer pays an amount X (in thousands) for a used car and then sells it for an amount Y. Suppose that X and Y have the joint density fxy (x, y) = for 0 < x < y < 6. The dealer's expected profit (in thousands) from the sale is equal toX and Y re random losses with joint density funcTion 2C forocn<00 500000 An Insurance losses pays the total of the twX9 losses to a maximum and oey< lo0, 100 policy on the payment of 100 the expectic fpagment the insurer will make on this policy.Let A > 0. Suppose (X, Y) has joint probability density function given, for all x, y E R, by 3. fx,y(x, y) = { Ay exp(-y(x+ A)), if a 2 0 and y > 0 0, otherwise. (i) Calculate the marginal density of Y. (ii) Let y > 0. Find the the conditional probability density function of X given Y = y. Interpret your answer. (iii) Calculate E[X|Y] and prove that E[X] = +oo. Hint: You can use without a proof that -dy = +oo. %3DRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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