5.3.([1] 5.5) The joint density of Y and Y2 is given by 0 < y2 < y1 <1 else f(y1.92) = {3 a) Find F (33) = P[Y...
5.4.([1] 5.6) The joint density function for Y1 and Y2 is f(y1,92) = {o 0 < y1 = 1,0 < y2 = 1 else a) what is P[Y1-Y2>0.5]? b) what is P[Y|Y2<0.5]?
Let Yı, Y, have the joint density S 2, 0 < y2 <yi <1 f(y1, y2) = 0, elsewhere. Use the method of transformation to derive the joint density function for U1 = Y/Y2,U2 = Y2, and then derive the marginal density of U1.
2. Let the random variables Y1 and Y, have joint density Ayſy22 - y2) 0<yi <1, 0 < y2 < 2 f(y1, y2) = { otherwise Stom.vn) = { isiml2 –») 05451,05 ms one a independent, amits your respon a) Are Y1 and Y2 independent? Justify your response. b) Find P(Y1Y2 < 0.5). on the
Consider two random variables with joint density fY1,Y2(y1,y2) =(2(1−y2) 0 ≤ y1 ≤ c,0 ≤ y2 ≤ c 0 otherwise (a) Find a value for c. (4 marks) (b) Derive the density function of Z = Y1Y2. (10 marks) . Consider two random variables with joint density fyiy(91, y2) = 2(1 - y2) 0<n<C,0<42 <c o otherwise (a) Find a value for c. (4 marks) (b) Derive the density function of Z=Y Y. (10 marks)
[15] 5. (X, Y) have joint density (22 + y?) 0<*<1 0<y<1 f(x, y) else find the marginals fx(x) and fy (y).
Let Y1, Y2 have the joint density f(y1,y2) = 4y1y2 for 0 ≤ y1,y2 ≤ 1 = 0 otherwise (a) (8 pts) Calculate Cov(Y1, Y2). (b) (3 pts) Are Y1 and Y2 are independent? Prove your answer rigorously. (c) (6 pts) Find the conditional mean E(Y2|Y1 = 1). 3
Q 3. The joint density of Yı, Y2 is given by e-4342 p(y1, y2) = - T, Y1 = 0,1, 2, ...; Y2 = 0, 1, 2, ... a. Find the marginal distribution of Yı. b. Find the conditional distribution of Y2 given that Y1 = yı. c. Determine if Yı and Y2 are independent - justify; you can use your result from b.
Let Y1 have the joint probability density function given by and Y2 k(1 y2), 0 s y1 y2 1, lo, = elsewhere. (a) Find the value of k that makes this a probability density function k = (b) Find P 1 (Round your answer to four decimal places.)
Is a joint density function? If yes, assume it is the joint density function of r.v.s X and Y , and compute the marginal densities of X and Y . f(r,y) = { " 0 <y<<11 , otherwise
7. Suppose that the joint density of X and Y is given by f(x,y) = e-ney, if 0 < x < f(z, y) = otherwise. Find P(X > 1|Y = y)