161-A3.2

Exercises

1. (3 points) Let $X,Y$ be independent and uniformly distributed random variables over the common alphabet $\{0,1\}$, and define a third random variable $Z$ as $Z=X\oplus Y$, where $\oplus$ is the XOR operation. Show that the three random variables are pairwise independent, but not mutually independent.
2. (3 points) Show that if $X$ is not a deterministic random variable, then $H(X)$ is strictly positive.
3. (2 points) Show that if $X\perp Z$, then $I(X;Y|Z)=I(X;Y,Z)$.
4. (2 points) For $n>3$, show that if $X_{1}\rightarrow X_{2}\rightarrow \cdots \rightarrow X_{n}$ forms a Markov chain, then $(X_{1},X_{2})\rightarrow (X_{2},X_{3})\rightarrow \cdots \rightarrow (X_{n-1},X_{n})$ also forms a Markov chain.