
PROBLEM 4 Let X be a continuous random variable with the following PDF 6x(1 - 1) if 0 <r<1 fx(x) = o.w. Suppose t...
Let X be a continuous random variable with the following PDF 6x(1 – x) if 0 < x < 1 fx(x) = 3 0.w. Suppose that we know Y | X = x ~ Geometric(x). Find the posterior density of X given Y = 2, i.e., fxY (2|2).
Let X and Y be two jointly continuous random variables with joint PDF xy0x, y < 1 fxy (x, y) O.W Find the MAP and ML estimates of X given Y = y
Let X be a continuous random variable with PDF fx(x)- 0 otherwise We know that given Xx, the random variable Y is uniformly distributed on [-x,x. 1. Find the joint PDF fx(x, y) 2. Find fyo). 3. Find P(IYI <x3)
Let X be a continuous random variable with PDF fx(x)- 0 otherwise We know that given Xx, the random variable Y is uniformly distributed on [-x,x. 1. Find the joint PDF fx(x, y) 2. Find fyo). 3. Find P(IYI
4. (20%) Let X be a continuous random variable with the following PDF Sce-4x 0<x fx(x) = to else where c is a positive constant. (a) (5%) Find c. (b) (5%) Find the CDF of X, Fx(x). (c) (5%) Find Prob{2<x<5} (d)(5%) Find E[X], and Var(X).
1. Let X be a random variable with variance ? > 0 and fx as a probability density function (pdf). The pdf is positive for all real numbers, that is fx(x) > 0. for all r ER Furthermore, the pdf fx is symmetric around zero, that is fx(x) = fx(-1), for all r ER Let y be the random variable given by Y = 4X2 +6X + with a,b,c E R. (i) For which values of a, b, and care...
Let X and Y be continuous random variables with joint pdf f(x,y) =fX (c(X + Y), 0 < y < x <1 otBerwise a. Find c. b. Find the joint pdf of S = Y and T = XY. c. Find the marginal pdf of T. 、
8. Let X and Y be a random variable with joint continuous pdf: f(x,y)- 0< y <1 0, otherwise a. b. c. Find the marginal PDF of X and Y Find the E(X) and Var(X) Find the P(X> Y)
Problem 5. Suppose that the continuous random variable X has the distribution fx(x), -00 <oo, which is symmetric about the value r 0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number. Hint: Use integration by parts
Let X and Y be random variables with joint PDF fx,y(x, y) = 2 for 0 < y < x < 1. Find Var(Y|X).
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...