prove Given two r.v’s X and Y over the same alphabet: (a) Show that the Kullback–Leibler...

prove

Given two r.v’s X and Y over the same alphabet:
(a) Show that the Kullback–Leibler distance D(P(X,Y) || P(X) P(Y)) = H(X) – H(X|Y)
(b) Show that the bounds of Mutual Information (MI) are 0 ≤ I(X:Y) ≤ min [ H(X) , H(Y) ]
with equality on the left if and only if X and Y are independent random variables, and
with equality on the right if and only if either Y essentially determines X, or X essentially
determines Y , or both.
(c) Show that the MI is symmetric, i.e. I(X;Y) = I(Y:X)

Expert Solution

orchestra answered 2 years ago

Given two r.v’s X and Y over the same alphabet: (a) Show that the Kullback–Leibler distance...

Given two r.v’s X and Y over the same alphabet: (a) Show that the Kullback–Leibler distance D(P(X,Y) || P(X) P(Y)) = H(X) – H(X|Y) (b) Show that the bounds of Mutual Information (MI) are 0 ≤ I(X:Y) ≤ min [ H(X) , H(Y) ] with equality on the left if and only if X and Y are independent random variables, and with equality on the right if and only if either Y essentially determines X, or X essentially determines Y...

Use Context-Free Pumping Lemma to prove that that following languages over the alphabet {'x', 'y', 'z'}...

Use Context-Free Pumping Lemma to prove that that following languages over the alphabet {'x', 'y', 'z'} are NOT context-free (a) {xjy2jzj : j > 0} (b) { xmynzk : m, n, k ≥ 0 and k = min(m,n) }

Given f(x,y) = 2 ; 0< x ≤ y < 1 a. Prove that f(x,y) is...

Given f(x,y) = 2 ; 0< x ≤ y < 1 a. Prove that f(x,y) is a joint pdf. b. Find the correlation coefficient of X and Y.

Given X + Y has a normal distribution, prove that X and Y each have a...

Given X + Y has a normal distribution, prove that X and Y each have a normal distribution if X and Y are iid random variables

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is...

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is a vector space over R. A polynomial f(x, y) is called degree d homogenous polynomial if the combined degree in x and y of each term is d. Let Vd be the set of degree d homogenous polynomials from R[x, y]. Is Vd a subspace of R[x, y]? Prove your answer.

Use two different ways to prove X Y + Z = (X + Z)(Y + Z)....

Use two different ways to prove X Y + Z = (X + Z)(Y + Z). a) Use pure algebraic way b) k-maps

1. Prove the language L is not regular, over the alphabet Σ = {a, b}. L...

1. Prove the language L is not regular, over the alphabet Σ = {a, b}. L = { aib2i : i > 0} 2) Prove the language M is not regular, over the alphabet Σ = {a, b}. M = { wwR : w is an element of Σ* i.e. w is any string, and wR means the string w written in reverse}. In other words, language M is even-length palindromes.

Consider a consumer with preferences over two goods (good x and good y) given by u...

Consider a consumer with preferences over two goods (good x and good y) given by u ( x , y ) = x ⋅ y. Given income of I and price of good yas $ P yper pound and price of good xgiven as $ P xper pound, the consumer chooses the optimal consumption bundle given as x ∗ = I 2 P x and y ∗ = I 2 P y . Given P x = $ 1per pound...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y < y^(−1) then x < −1.

2. Prove the following properties. (b) Prove that x + ¯ xy = x + y.

2. Prove the following properties.(b) Prove that x + ¯ xy = x + y.3. Consider the following Boolean function: F = x¯ y + xy¯ z + xyz(a) Draw a circuit diagram to obtain the output F. (b) Use the Boolean algebra theorems to simplify the output function F into the minimum number of input literals.

Question