Exercise 6. Let (X, Y) be uniform on the unit ball, i.e. it has density if...
55. Let X and Y be jointly continuous random variables with joint density function fx.y(x,y) be-3y -a < x < 2a, 0) < y < 00, otherwise. Assume that E[XY] = 1/6. (a) Find a and b such that fx,y is a valid joint pdf. You may want to use the fact that du = 1. u 6. и е (b) Find the conditional pdf of X given Y = y where 0 <y < . (c) Find Cov(X,Y). (d)...
Do only (b). Use convolution. Exercise 7.20. Let X have density fx(x) = 2x for 0 < x < 1 and let Y be uniform on the interval (1,2). Assume X and Y independent. Give the joint density function of (X. Y). Calculate P(Y - X2 Find the density function of X + Y. (a) (b)
the answer should be 1/2y^2 3. Suppose the joint density of X and Y is defined by if 0<r<y< 1 f(x,y)= elsewhere. What is E (X2Y = ) ?
2. Let R be the region R = {(X,Y)|X2 + y2 < 2} and let (X,Y) be a pair of random variables that is distributed uniformly on this region. That is fx,y(x, y) is constant in this region and 0 elsewhere. State the sample space and find the probability that the random variable x2 + y2 is less than 1, P[X2 +Y? < 1].
Question-2 Consider the joint uniform density function C for 22 + y2 < 4, f(x,y) 0 otherwise. What is the value of c? 0 What is P(X<0)? What is P(X <0, Y <0)? What is f( x | y=1)?
A point (X, Y ) in the Cartesian plane is uniformly distributed within the unit circle if X and Y have joint density Find the marginal densities fX and fY and state whether X and Y are independent or not. Provide a mathematical justification for your answer. 1, 22 + y2 <1, f(x, y) = { 1 0, otherwise.
Let the random variables X, Y with joint probability density function (pdf) fxy(z, y) = cry, where 0 < y < z < 2. (a) Find the value of c that makes fx.y (a, y) a valid pdf. (b) Calculate the marginal density functions for X and Y (c) Find the conditional density function of Y X (d) Calculate E(X) and EYIX) (e Show whether X. Y are independent or not.
Let f(x,y) = exp(-x) be a probability density function over the plane. Find the probabilities: Parta)P( X2 + y2 <a), a > 0, Part b)P(x2 + y2 <a), a > 0.
(1 point) Let Yı, Y2, ..., Yn be a random sample from the probability density function f(yla) = |aya-2/5° f(y ) 0 <y< 5 otherwise 0 for > -1. Find an estimator for a using the method of moments.
. 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)