1. [10 marks) Let X, and X, have joint density function f(0,0%) = cca 110, x2...
EXERCISE (x2+1), where . < 1) A random variable X has the density function f(x)= a) Find the value of the constant C b) Find the probability that X lies between 1/3 and 1
The joint density of random variables X1, X2 is given by fx1,x2 (x1, 2)= 6x1, for 0 < xı < 1, 0 2 <1 - r Let Y X1X2. Find the joint density of Yi and Y2 Х1, Y?
1. Let X1, X2, X3 be continuous random variables with joint probability density function 00 < Xi < 00,i=1,2,3 Consider the transformation U-X1, V = X , W-XY + X + X (a) Find the joint pdf (probability density function) of U, V and W. (b) Find the marginal pdf of U, and hence find E(U) and Var(U) (c) Find the marginal pdf of W, and hence find E(W) and Var(W) (d) Find the conditional pdf of U given Ww,...
2. Let Xi and X2 be two continuous random variables having the joint probability density 1X2 , for 0, elsewhere. If Y-X? and Y XX find a. the joint pdf of Yǐ and Y, g(n,n), b. the P(Y> Y), c, the marginal pdfs gi (m) and 92(h), d. the conditional pdf h(galn), and e, the E(YSM-m) and E(%)Yi = 1/2).
4. Suppose two random variables X and Y has the following joint density function Cry, 22 Sy<1, f(x,y) = { 0, otherwise. (a) Find the constant C. (b) Find E(Y|X = 1/2). 5. Suppose X1, X2, ..., are i.i.d. random variables coming from the N(0,0%) population. (a) Determine the mean and variance for X. (b) Show that va bos (x2) – 1o60*) $ (0.2).
Let X- (Xi, X2,X3) be an absolutely continuous random vector with the joint probability density function elsewhere. Calculate 1. the probability of the event A -(Xs 3. the probability density function xx (,s) of the (XX)-marginal 4. the probability density function fx, () of the Xi-marginal, and the probability density function fx (r3) of the X3-marginal 5. Are Xi and X independent random variables? 6. E(Xi) and Var(X) 8. the covariance cov(Xi, X3) of Xi and X,3 9. Which elements...
3. Let (X1, X2) have the joint p.d.f 1 if 0 <1,0 < <1 f(1, ) else Find P(X1X2 < 0.5)
5. (10 pts )The random variables X and Y have joint density function 1 f(x,y) x2 + y2 <1. 3 7T Compute the joint density function of R= x2 + y2 and = tan-'(Y/X).
If the probability density function of X is given by n2 for 1<x< 2 fx ) = 10 elsewhere (a) Find, E[X], E[X2], and E[X3] (b) Use your answer to part (a) to find E[X3 + 3X2 - 2x + 5)
2. (10 marks) Let X, and X, be two random variables with joint pdf 3.1 0 < x <3 <1; xix,( 22) - Yo elsewhere. a) Are X, and X, independent? If not, find E(X,X2). b) Are X, and X, correlated? Find Cou(X1, X2).