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## Homework Statement

Let ##X## and ##Y## be two independent integer-valued random variables. Let ##X## be uniformly distributed over ##\left\{1,2,...,8\right\}##, and let ##\text{Pr}\left\{Y=k\right\} =2^{-k},~k=1,2,3,...##

(a) Find ##H(X)##.

(b) Find ##H(Y)##.

(c) Find ##H(X+Y,X-Y)##.

## Homework Equations

I am confused about part (c). I have found the answers to (a) and (b), they are obviously 3 bits and 2 bits, respectively. However, the solution I get for (c) does not match the answer. The answer to (c) is apparently 5 bits.

## The Attempt at a Solution

I argue that ##Z=X+Y## and ##W=X-Y##. Thus, I create the vectors ##\mathbf{u} = [Z,W]^T## and ##\mathbf{v}=[X,Y]^T## and write them as a linear transformation of each other as

##\mathbf{u}=\begin{bmatrix}1&1 \\ 1&-1 \end{bmatrix}\mathbf{v}=\mathbf{M}\mathbf{v}##.

Therefore, ##H(\mathbf{u})=H(X+Y,X-Y)=H(\mathbf{v})+\log_2\lvert\text{det}\left(\mathbf{M}\right)\rvert##. I then have

##\log_2\lvert\text{det}\left(\mathbf{M}\right)\rvert=1## bit

##H(\mathbf{v})=H(X)+H(Y|X)=H(X)+H(Y)=5## bits (since ##Y## is independent of ##X##).

This leaves me with the answer for (c) to be 6 bits.

Edit: Unless the formula I am using with log-det is only for continuous and not discrete distributions.

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