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 1 year 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...
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...
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....
Given two functions, M(x, y) and N(x, y), suppose that (∂N/∂x − ∂M/∂y)/(M − N) is...
Given two functions, M(x, y) and N(x, y), suppose that (∂N/∂x −
∂M/∂y)/(M − N)
is a function of x + y. That is, let f(t) be a function such
that
f(x + y) = (∂N/∂x − ∂M/∂y)/(M − N)
Assume that you can solve the differential equation
M dx + N dy = 0
by multiplying by an integrating factor μ that makes it exact
and that it can also be
written as a function of x + y,...
Let L be the set of all languages over alphabet {0}. Show that L is uncountable,...
Let L be the set of all languages over alphabet {0}. Show that L
is uncountable, using a proof by diagonalization.
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