show all work please TTIVITICUL ycuci aunty luiICUIUIT 1. x and y follow this distribution: 2...
(п-1)S? for the conditional 1-3) Show that the moment generating function(MGF) of distribution of 2,given X is (n-1)S2 | X (1-2 -(n-l)/2 ,1 < 2 E expt Hint: Notice that g,,, is a pdf That is, 7 1- "ppxp )./ (n-1)S2 X Еl exp| t in a multi-integral form using the conditional pdf of Express X2,, given X Then try to consider the integrand as another joint pdf times a constant. Then the answer will be the constant. Hint
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STAT 140 Suppose that X have a gamma distribution with parameters a = 2 and θ= 3, and suppose that the conditional distribution of Y given X=x, is uniform between 0 and x. (1) Find E(Y) and Var(Y). (2) Find the Moment Generating Function (MGF) of Y. What is the distribution of Y?
1. Show that X and Y are independent if and only if the conditional distribution of X given Y = y is the same as the marginal distribution of X for all y. When X and Y are independent show that E(X Y = y) = EX.
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find and sketch the marginal pdf fY(y)
The Joint distribution function for two rondom variables X and Y is Exy(x, y) = u(x)u(y) [l-e ax cara e acx+y)] where azo Find and sketch the marginal Pdf Fy (4)
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y have the moment generating function My (t) i. Show that the moment generating function of Z = aY b is e*My(at) for non-zero constants a and b ii. Use the result to write down the moment generating function of W 1- 2X if X Gamma(a, B)
(a) If var[X o2 for each Xi (i...
Let the conditional probability distribution of Y given π be elsewhere In this problem we will assume that π is a random variable and that the marginal distribution of π has a probability density function given by: f(n) = 0 elsewhere (a) Find the joint probability density function of Y and π, that is f(y, π). Please find the marginal probability distribution of Y, ). (c) Find the conditional distribution of f( y). (d) What is the mean and variance...
(n-1)S for the conditional 1-3) Show that the moment generating function(MGF) of distribution of2,A given X is ,(n-1)SX (1-2) (2,1 1 -(n-l)/2 E exp t 2 Hint: Notice that , is a pdf. That is, ] 77 (n-1)S | X E exp .2 in a multi-integral form using the conditional pdf of Express X2, given X. Then try to consider the integrand as another joint pdf times a constant. Then the answer will be the constant [Hint] [Hint 2] 22-1...
9. Let a random variable X follow the distribution with pdf f(z)=(0 otherwise (a) Find the moment generating function for X (b) Use the moment generating function to 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.