Due on Oct 24 (20 points)

1. (6 points) For a group of students with weight \(W\) and height \(H\) are jointly normally distributed with mean of 50 kg and 160 cm and a covariance matrices of \(E\left[\begin{matrix} (W-\bar{W})^2 & (W-\bar{W})(H-\bar{H}) \\ (H-\bar{H})(W-\bar{W}) & (H-\bar{H})^2 \end{matrix} \right]=\begin{pmatrix} 80 & 30 \\ 30 & 140 \end{pmatrix}\).

   1. Compute \(h(W)\)

   2. Compute \(h(W|H)\)

   3. how many bits on average will be needed per sample to store a student's weight with precision of 0.1 if his height is known?

2. (14 points) Consider Problem 1 in last HW. Denote \(D\) as the actual dice we drew, and \(X_1, X_2, X_3\) are the respective outcomes of three tosses. Compute

   1. \(H(X_1|D)\)

   2. \(H(D|X_1)\)

   3. \(H(D|X_1,X_2)\)

   4. \(I(X_1;X_2|D)\)

   5. \(I(X_1;D)\)

   6. \(I(X_1;D|X_2)\)

   7. \(I(X_1;X_2)\)