Question
In: Statistics and Probability
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 , or both.
(c) Show that the MI is symmetric, i.e. I(X;Y) = I(Y:X)
Solutions
orchestra answered 2 years agoRelated Solutions
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...
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) }
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...
Problem 3 The function distance returns the distance between two strings X and Y. Its running...
Problem 3
The function distance returns the distance between two
strings X and Y. Its running time is exponential. Explain why.
(Note: make sure you understand what X[0: -1] is.)
#
# Input: X, Y, which are strings
# Output: a number, which is the distance between the two strings
#
def distance(X, Y):
if len(X)==0:
return len(Y)
if len(Y)==0:
return len(X)
if X[-1]==Y[-1]:
return distance(X[0:-1], Y[0:-1])
else:
a = distance(X[0:-1], Y[0:-1])
b = distance(X, Y[0:-1])
c = distance(X[0:-1], Y)...
4. Two companies, X and Y, are competitors in the same industry, but company X is...
4. Two companies, X and Y, are competitors in the same industry,
but company X is more efficient at using its resources than company
Y.
a. If the MRTS is the same in the production processes of both
companies, show and discuss how the isoquants will be different for
company X than they are for company Y.
b. Both companies can hire labor at the same wage, w, but
company X is able to pay a lower rent per unit...
Two companies, X and Y, are competitors in the same industry, but company X is more...
Two companies, X and Y, are competitors in the same industry,
but company X is more efficient at using its resources than company
Y
. a. If the MRTS is the same in the production processes of both
companies, show and discuss how the isoquants will be different for
company X than they are for company Y.
b. Both companies can hire labor at the same wage, w, but
company X is able to pay a lower rent per unit...
Two companies, X and Y, are competitors in the same industry, but company X is more...
Two companies, X and Y, are
competitors in the same industry, but company X is more efficient
at using its resources than company Y.
a. If the MRTS is
the same in the production processes of both companies, show and
discuss how the isoquants will be different for company X than they
are for company Y.
b. Both companies
can hire labor at the same wage, w, but company X is able to pay a
lower rent per unit on...
4.A palindrome is a nonempty string over some alphabet that reads the same forward and backward....
4.A palindrome is a
nonempty string over some alphabet that reads the same forward and
backward. Examples of palindromes are all strings of length 1,
civic, racecar,
abba,…
Give a DP algorithm to find the longest palindrome that
is a subsequence of a given input string. What is the running time
of your algorithm? (40 points).
•Examples:
•Given the input “character”, your algorithm should return
“carac”.
•Given the input “TTATGTGAT”, your algorithm should return
“TATGTAT”
Python Code only
Compute the correlation between X and the transformed Y values. Distance jumped in inches (X) Distance...
Compute the correlation between X and the transformed Y
values.
Distance jumped in inches (X)
Distance of discus throw in inches (Y)
3
12
1
36
4
60
1
12
3
48
5
12
1
24
A consumes two goods, x and y. A ’s utility function is given by u(x, y)...
A consumes two goods, x and y. A ’s utility function is given
by u(x, y) = x 1/2y 1/2
The price of x is p and the price of y is 1. A has an income
of M.
(a) Derive A ’s demand functions for x and y.
(b) Suppose M = 72 and p falls from 9 to 4. Calculate the
income and substitution effects of the price change.
(c) Calculate the compensating variation of the price
change....
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