Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x, 0<x<1 (zero otherwise), and g(y) = 3y2, 0<y<1 (zero otherwise). Let U = min(X,Y), and V = max(X,Y). Find the joint pdf of U and V.
Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x,...
sity functions. Exercise 6.46. Let X, Y be independent random variables with density functions fx and fy. Let T- min(X, Y) and V max(X, Y). Use the joint density function fr v from equation (6.31) to compute the marginal density functions fr of T and fv of V
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
Let X and Y be jointly continuous random variables with joint probability density given by f(x, y) = 12/5(2x − x2 − xy) for 0 < x < 1, 0 < y < 1 0 otherwise (a) Find the marginal densities for X and Y . (b) Find the conditional density for X given Y = y and the conditional density for Y given X = x. (c) Compute the probability P(1/2 < X < 1|Y =1/4). (d) Determine whether...
3.5. Suppose that X and Tare independent, continuous random variables and that U-X+1. Denote their probability density functions by f(x), g(y) and h(u) and the corresponding cumulative probability functions by F(x), G(2) and H(u) respectively. Then For a fixed value of I, say T-y,this probability is F(u-), and the probability that I will lie in the range y to y+dy is g()dy. Hence the probability that Usu and that simultaneously Y lies between y and y+dy is F(u-)go)dy and so...
2. Let X and Y be continuous random variables with joint probability density function fx,y(x,y) 0, otherwise (a) Compute the value of k that will make f(x, y) a legitimate joint probability density function. Use f(x.y) with that value of k as the joint probability density function of X, Y in parts (b),(c).(d),(e (b) Find the probability density functions of X and Y. (c) Find the expected values of X, Y and XY (d) Compute the covariance Cov(X,Y) of X...
Let X and Y be two random variables with the joint probability density function: f(x,y) = cxy, for 0 < x < 3 and 0 < y < x a) Determine the value of the constant c such that the expression above is valid. b) Find the marginal density functions for X and Y. c) Are X and Y independent random variables? d) Find E[X].
Suppose X and Y are jointly continuous random variables with joint density function Let U = 2X − Y and V = 2X + Y (i). What is the joint density function of U and V ? (ii). Calculate Var(U |V ). 1. Suppose X and Y are jointly continuous random variables with join density function Lei otherwise Let U = 2X-Y and V = 2X + y (i). What is the joint density function of U and V? (ii)....
Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0 otherwise Show that the joint density function of U = 3(X-Y) and V = Y is otherwise, where A is a region of the (u, v) plane to be determined. Deduce that U has the bilateral exponential distribution with density function fu (11) te-lul foru R. Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0...
8), Let X and Y be continuous random variables with joint density function f(x,y)-4xy for 0 < x < y < 1 Otherwise What is the joint density of U and V Y
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi , . . . , X,.), V=min(X1, ,X,). (a) Find the distribution function and the density function of U and of V (b) Show that the joint density function of U and V is fe,y(u, u)= n(n-1)/(u)/(v)[F(v)-F(u)]n-1, ifu < u. (7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi...