A random variable x = {0, 1,2,3} is selected by flipping a bent coin with bias f to determine whether the outcome is in {0, 1} or {2,3}; then either flipping a second bent coin with bias g or a third bent coin with bias h respectively. Write down the probability distribution of x. Use the decomposability of the entropy (2.44) to find the entropy of X. [Notice how compact an expression is obtained if you make use of the binary entropy function H₂(x), compared with writing out the four-term entropy explicitly1 Find the derivative of H(X) with respect
A random variable x = {0, 1,2,3} is selected by flipping a bent coin with bias f to determine whether the outcome is in {0, 1} or {2,3}; then either flipping a second bent coin with bias g or a third bent coin with bias h respectively. Write down the probability distribution of x. Use the decomposability of the entropy (2.44) to find the entropy of X. [Notice how compact an expression is obtained if you make use of the binary entropy function H₂(x), compared with writing out the four-term entropy explicitly1 Find the derivative of H(X) with respect
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![A random variable x {0, 1, 2, 3} is selected by flipping
a bent coin with bias f to determine whether the outcome is in {0, 1} or
{2,3}; then either flipping a second bent coin with bias g or a third bent
coin with bias h respectively. Write down the probability distribution
of x. Use the decomposability of the entropy (2.44) to find the entropy
of X. [Notice how compact an expression is obtained if you make use
of the binary entropy function H₂(x), compared with writing out the
four-term entropy explicitly.] Find the derivative of H(X) with respect
to f. [Hint: dH₂(x)/dx = log((1 − x)/x).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6aa3972b-42fd-4773-b460-1372118ea0ee%2F629f3228-462f-40e6-aa08-5a4c4179254b%2Ffkrvvuq_processed.png&w=3840&q=75)
Transcribed Image Text:A random variable x {0, 1, 2, 3} is selected by flipping
a bent coin with bias f to determine whether the outcome is in {0, 1} or
{2,3}; then either flipping a second bent coin with bias g or a third bent
coin with bias h respectively. Write down the probability distribution
of x. Use the decomposability of the entropy (2.44) to find the entropy
of X. [Notice how compact an expression is obtained if you make use
of the binary entropy function H₂(x), compared with writing out the
four-term entropy explicitly.] Find the derivative of H(X) with respect
to f. [Hint: dH₂(x)/dx = log((1 − x)/x).]
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