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Assume suitable data wherever necessary The relative entropy is : measure of the distance two distributions statistics The that when For example we knew distribution average Definition two probability mass xex Relative entropy always nonnegative is zero and only if p = q. However, it is not a true distance between distributions since not symmetric and does not satisfy the triangle inequality. Nonetheless often useful think relative "distance" between a)Specify the value of the following If there symbol Y such that i) p(s)-0 q(x)-0 q(x)>0 iii)(p(x)>0 g(x)-0 b) Let and q(0) q(1)= find the values D(pliq) and D(qlip) 2. H(X1. X1) - lf the average conditional mutual information of random variables x and given Zis defined b 1(X:Y\Z)= H(XZ)-H(XIY.Z) 3. Prove that I(X1.X2 i=] Differential Entropy and Conditional Entropy for continuous random variables (Extending defisations attached) The Differential Entropy of continuous random variable Xi defined as The Average Conditional Entropy of continuous random variable X given Y is defined as The Average Mutual Information between two continuous random variables x and Y Let and be random variables with joint probability density function (pdf) p(x.y) and marginal pdfs p(x) and p(y) The Average Mutual Information between two continuous random variables and Y defined as I(X:Y) log( p(y)x)p(x)((p(x)p(y))) dx dy The average mutual information can be expressed as Explain why the definitions entropy. average conditional entropy. average mutual information can carried over from discrete random variables to randorn variables, but the concept and physical interpretation cannot? b) variables with joint PDF otherwise Find the marginal PDFs fx(z) and fx(y) Suppose Discrete Memoryless Source (DMS) outputs symbol every seconds Each symbol finite set of symbols, occurring with What bits source symbol Explain how d) Wheni represent 26 letters the English number bits required to letter represented using the c) Write will segment that Same except which : i) p(x) = 0 , q(x) = 0 ii) p(x) = 0 , q(x) > 0 iii) p(x) > 0 , q(x) = 0 Find D(p||q) and D(q||p). Also, find the values of D(p||q) and D(q||p) when r = s. : c) Write a code to call functions which will play the role of tables for z, t, chi-square and F distributions, namely, for z, write a code segment that will take: (a) z0 as input and give P(z<z0) as output; (b) alpha as input and give z0 as output where P(z<z0)=alpha. Same for the rest, except it will take degree(s) of freedom (dof) also as input. : 4 a) Explain why the definitions of entropy, average conditional entropy, average mutual information can be carried over from discrete random variables to continuous random variables, but the concept and physical interpretations cannot? : c) What is the entropy H(X) of this DMS in bits per source symbol ? Explain how this H(X) < = log2L d) When is this H(X) = log2L ? Suppose we wish to represent 26 letters of the English alphabet in bits, what is the minimum number of bits required to uniquely represent each of the letters (such that every letter is represented using the same number of bits) ? c) Write a code in python or in any programming language: to call functions which will play the role of tables for z, t, chi-square and F distributions,namely, for z, write a code segment that will take: (a) z0 as input and give P(z<z0) as output; (b) alpha as input and give z0 as output where P(z<z0)=alpha. Same for the rest, except it will take degree(s) of freedom (dof) also as input. (This part is related to sampling distributions of statistics, which should be implemented on personal computer.) ***

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4) C.
Z table
i) When z0 as input>>>>>>>>>>

from scipy.stats import norm
z0 = float(input("Z0="))
value = norm.cdf(z0)
print(value)


ii) When alpha as the input>>>>>>>>

from scipy.stats import norm
alpha = float(input("Alpha="))
value = norm.ppf(alpha)
print(value)


t Table
i) When z0 as input>>>>>>>>>

from scipy.stats import t
Z0 = float(input("Z0="))
df =int(input("Degree of Freedom="))
value = t.cdf(Z0, df)
print (value)

ii) When alpha as input>>>>>>

from scipy.stats import t
alpha = float(input("Alpha="))
df =int(input("Degree of Freedom="))
value = t.ppf(alpha, df)
print (value)


Chi Square Distribution
i) When z0 as input>>>>>>>>>...
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