2Chain rule.Recall that the chain rule for entropy for two random variables is: H(X,Y) = H(X)+Η (Y | Χ).(a) Prove that for three random variables X, Y and Z, a conditional chain rule holds:H(X, Z|Y) = H(X|Y)+ H(Z|X,Y)(b) Starting from the chain rule for entropy, prove that for four random variables X,Y, Z and U, the following identity holds:I(X,Y; Z\U) = I(X; Z|U)+I(Y; Z|X,U)

Given that the chain rule for entropy for two random variables, HX,Y=HX+HY/X…

Answered: Recall that the chain rule for entropy…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose that X is a Gamma distributed random variable with parameters a and 2. Additionally, Y is uniform on [0,1] and that X and Y are independent. a) Compute the variance of Z = XY. b) Compute the moment generating function (mgf) of W = X + Y.
Prove that the maximal entropy of a discrete random variable is logan (n being the number of possible values of the random variable) and is attained for P₁ = P2 = = P₁ = 1/n.
Q2) The joint probability function of two discrete random variables X and Y is given by fx,y) = c(2x+y),where x and y can assume all integers such that 0 sIS2,0s ys3,and fx,y) = 0 otherwise. (a) Find the value of the constant c (b) Cov (X, Y), (c) p.
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Consider the pair of random variables ( X , Y ) with joint PDF f X , Y ( x , y ) = 2 for 0 ≤ x ≤ y ≤ 1 and f X , Y ( x , y ) = 0 otherwise. (In particular, note that the joint PDF is only nonzero when x ≤ y .) (a) Draw a picture of the range of ( X , Y ) in the x y -plane. (b) Describe the conditional distribution of X given Y = y , for 0 ≤ y ≤ 1 . (c) What is the marginal PDF f X of X ? (d) Are X and Y independent?
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Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a) Prove that X^2 and Y are independent. Note that by symmetry, it will also follow that Y^2 and X are independent. (b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).
The joint probability function of two discrete random variables X and Y is given by Ax,y) = c(2x+y), where x and y can assume all integers such that 0< x
The joint pdf of the random variables X and Y is given by { cx, x > 0, y > 0, 1 < x + y < 2, f (x, y) 0, elsewhere, for some constant c. (a) Draw the support of (X, Y). (b) Show that c= = 6/7.
Let X₁, X₂ be independent and identically distributed random variables following a geometric distribution with parameter p € (0, 1), that is, P(X₁ = k) = pk-1(1-p) for k € N. a) Derive a formula for P(X₁2 k) for ke N. b) Determine the cumulative distribution function of Y = min(X₁, X₂). c) Let Z= max(X₁, X₂). Are Y and Z independent? Justify your answer.
Example 18.2 For the probability density of a system of random variables (X, Y): f(x, y) = 0.5 sin (x + y) (0 < x < 7,0 << 7), 2' determine (a) the distribution function of the system, (b) the expectations of X and Y, (c) the covariance matrix.
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