5. Let X have the uniform distribution U(0, 1), and let the conditional distribution of Y, given X = x, be U(0, x).
Find P(X + Y ≥ 1).
5. Let X have the uniform distribution U(0, 1), and let the conditional distribution of Y,...
Question 15: Let Π is distributed as Uniform(0, 1) and
the conditional distribution of X given Π = π is Bernoulli (π).
Find the conditional distribution of Π given X = x.
Question 15: Let Π is distributed as Uniform(0, 1) and the conditional distribution of X given 11 = π is Bernoulli (π). Find the conditional distribution of Π given X = x
(c) (20 pts.) Let X have a uniform distribution U(0, 2) and let the considiton; distribution of Y given X = x be U(0, x3) i. Determine f (x, y). Make sure to describe the support of f. ii. Calculate fy (y) iii. Find E(Y).
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
Let X be a uniform(0, 1) random variable and let Y be uniform(1,2) with X and Y being independent. Let U = X/Y and V = X. (a) Find the joint distribution of U and V . (b) Find the marginal distributions of U.
Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given that Y -y is uniform on (0, y). Find E[X] and Var(X).
Suppose Y is uniformly distributed on (0, 1), and that the conditional distribution of X given that Y -y is uniform on (0, y). Find E[X] and Var(X).
10. Let X and Y have a discrete joint distribution with if (x,y) = (-1,1) P(X = 2, Y = y) = { = ; if x=y=0 = 0, elsewhere Find (a) the conditional distribution of Y given X = -1. (b) show that X and Y are uncorrelated but not independent. (C) Find the marginal distributions of X and Y.
Let U ~uniform(0,1). Let Y =−ln(1−U). hint: If FX (x) = FY (y) and supports x,y ∈ D, X and Y have the same distribution. Find FY (y) and fY (y). Now, it should be straight forward that Y follows distribution with parameter_____________-
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y has an exponential distribution with parameter = 1. (Remember to justify all of your answers.) (a) What is the joint distribution of X and Y? (b) What is P{(x > 0.25) U (Y > 0.25)}? (c) What is the conditional distribution of X. given that Y - 3? (d) What is Var(Y - E[2X] + 3)? (e) What is...
Let X be an uniform distribution between 0 and 1, Y be an uniform distribution between -5 and 3, and they are independent. Calculate the pdf, expectation, and the variance of the followings 1. 4X