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Question: suppose the joint probability distribution of two binary random variables...

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Suppose the joint probability distribution of two binary random variables X and Y are given as follows.

x/y 1 2
0 3/10 0
1 4/10 3/10

X goes along side as 0 and 1, Y goes along top as 1 and 2.

a) Show the marginal distribution of X.

b) Find entropy H(Y ).

c) Find conditional entropy H(X|Y ) and H(Y |X).

d) Find mutual information I(X; Y ).

e) Find joint entropy H(X, Y ).

f) Suppose X and Y are independent. Show that H(X|Y ) = H(X).

g) Suppose X and Y are independent. Show that H(X, Y ) = H(X) + H(Y ).

h) Show that I(X; X) = H(X).

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