Question 2(c)Let 𝑋1 → 𝑋2 → 𝑋3 → · · · → 𝑋𝑛 form a Markov chain in this order. Thus, the joint probabilityof 𝑋1, . . . , 𝑋𝑛 are given by𝑝(𝑥1, 𝑥 − 2, . . . , 𝑥𝑛) = 𝑝(𝑥𝑛 |𝑥𝑛−1) 𝑝(𝑥𝑛−1 |𝑥𝑛−2) · · · 𝑝(𝑥2 |𝑥1) 𝑝(𝑥1).1. Express 𝐼 (𝑋1; 𝑋2, . . . , 𝑋𝑛) in terms of its entropy and conditional entropy [2 Marks]2. Simplify the entropy expression you derived above to reduce 𝐼 (𝑋1; 𝑋2, . . . , 𝑋𝑛) to itssimplest form. Please do not use other methods to simplify, you will not receive anymarks.

1. The mutual information I(X1; X2, ..., Xn) can be expressed in terms of entropy and conditional entropy as follows:

I(X1; X2, ..., Xn) = H(X1) - H(X1|X2, ..., Xn)

Here, H(X1) is the entropy of X1 and H(X1|X2, ..., Xn) is the conditional entropy of X1 given X2 through Xn.

2. To simplify the entropy expression, we need to use the chain rule for conditional entropy. The chain rule states that the conditional entropy H(X1|X2, ..., Xn) can be written as a sum of conditional entropies:

H(X1|X2, ..., Xn) = H(X1|X2) + H(X2|X3) + ... + H(Xn-1|Xn)

Substituting this into the expression for mutual information gives:

I(X1; X2, ..., Xn) = H(X1) - (H(X1|X2) + H(X2|X3) + ... + H(Xn-1|Xn))

This is the simplest form of the expression for mutual information in terms of entropy and conditional entropy.