Exercise 3-6.1 Two random variables X and Y have a joint probability density function of the...
The joint probability density function of the random variables X, Y, and Z is (e-(x+y+z) f(x, y, z) 0 < x, 0 < y, 0 <z elsewhere (a) (3 pts) Verify that the joint density function is a valid density function. (b) (3 pts) Find the joint marginal density function of X and Y alone (by integrating over 2). (C) (4 pts) Find the marginal density functions for X and Y. (d) (3 pts) What are P(1 < X <...
. Let X and Y be the proportion of two random variables with joint probability density function f(x, y)o, elsewhere. (a) Find P(X < 3|Y= 2). (b) Are X and Y independent? Why? (c) Find E(Y/X)
. Let X and Y be the proportion of two random variables with joint probability density function f(x, y)o, elsewhere. (a) Find P(X < 3|Y= 2). (b) Are X and Y independent? Why? (c) Find E(Y/X)
(1 point) 3. Let X and Y be random variables with a joint probability density function f(z, y)e (a)Find the marginal distribution functions of X and Y, respectively. i.e. Find f(z) and f(y) f(x)- elsewhere (b) Identify the distribution of Y. What is the E(Y) and SD(Y) E(Y)- (c) Are X and Y independent random variables? Show why, or why not (d) Find P(1 X 2|Y 1) E SD(Y)-
3. Consider two random variables X and Y with the joint probability density (a)o elsewhere which is the sane asin Question I. Now let Z = XY 2 and U = X be a joint transformation of (X, Y). (a) Find the support of (Z, U) (b) Find the inverse transformation (c) Find the Jacobian of the inverse transformation. (d) Find the joint pdf of (Z, U) (e) Find the pdf of Z XY from the joint pdf of (Z,...
3. Suppose that two continuous random variables X and Y have a joint probability density function given as f(x,y)- a) Find the value ofAK K(x-3)y, -2sxs 3, and 4s ys6 elsewhere
. Let X and Y be the proportion of two random variables with joint probability density function f(r, y) e-*, 0, if, 0 < y < x < oo, elsewhere. a) Find P(Xc3.y-2). b) Are X and Y independent? Why? c) Find E(Y/X)
(45) Two random variables X and Y have the joint probability density | 2, 0sxs1 and 0 s ys1 and x + y21 fxY (x, y) = 0, elsewhere Answer each of these independent questions about X, Y, carefully indicating the domain of all functions where needed. Parts a). - i). are 5 points each. a). Find E(Z), where Z is a new random variable defined by Z = XY b). A is the event {X >0.75}. Find P(A). c)....
Consider the following joint probability density function of the random variables X and Y : 3x−y , 1 < x < 3, 1 < y < 2, f(x, y) = 9 0, elsewhere. (a) Find the marginal density functions of X and Y . (b) Are X and Y independent? (c) Find P(X > 2).
3. Let the random variables X and Y have the joint probability density function 0 y 1, 0 x < y fxy(x, y)y otherwise (a) Compute the joint expectation E(XY) (b) Compute the marginal expectations E(X) and E (Y) (c) Compute the covariance Cov(X, Y)