A joint law has rows for X and columns for Y. The rows are 1/4, 0, 0, 1/4, then 0, 1/8, 0, 1/8, then 0, 0, 1/8, 1/8. What is I(X; Y) in bits? Give a fraction.

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Reading a joint table

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A joint law has rows for $X$ and columns for $Y$ . The rows are $\frac{1}{4}, 0, 0, \frac{1}{4}$ , then $0, \frac{1}{8}, 0, \frac{1}{8}$ , then $0, 0, \frac{1}{8}, \frac{1}{8}$ . What is $I (X; Y)$ in bits? Give a fraction.

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Hint

Use $I (X; Y) = H (X) + H (Y) - H (X, Y)$ .

Solution

Row totals are $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{4}$ , so $H (X) = \frac{1}{2} \times 1 + \frac{1}{4} \times 2 + \frac{1}{4} \times 2 = \frac{3}{2}$ .
Column totals are $\frac{1}{4}$ , $\frac{1}{8}$ , $\frac{1}{8}$ , $\frac{1}{2}$ , so $H (Y) = \frac{1}{4} \times 2 + \frac{1}{8} \times 3 + \frac{1}{8} \times 3 + \frac{1}{2} \times 1 = \frac{7}{4}$ .
The $6$ cells give $H (X, Y) = \frac{2}{4} \times 2 + \frac{4}{8} \times 3 = \frac{5}{2}$ .
So $I (X; Y) = \frac{3}{2} + \frac{7}{4} - \frac{5}{2} = \frac{3}{4}$ .

Answer: $\frac{3}{4}$

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