Question

X and Y are Bernoulli random variables taking values in \( \{0,1\} \). The joint probability mass function of the random variables is given by:
P(X = 0, Y = 0) = 0.06, \quad P(X = 0, Y = 1) = 0.14, \quad P(X = 1, Y = 0) = 0.24, \quad P(X = 1, Y = 1) = 0.56. The mutual information \( I(X; Y) \) is (rounded off to two decimal places).

Hint:

Mutual information quantifies the amount of information shared between two random variables. If the mutual information is zero, the variables are independent.

Answer 1

Correct Answer: 0

The mutual information \( I(X; Y) \) is given by: \[ I(X; Y) = \sum_{x, y} P(x, y) \log_2 \frac{P(x, y)}{P(x)P(y)} \] Step 1: Compute the marginal probabilities
The marginal probabilities are given by summing over the appropriate values of \( Y \) and \( X \): \[ P(X = 0) = P(X = 0, Y = 0) + P(X = 0, Y = 1) = 0.06 + 0.14 = 0.20 \] \[ P(X = 1) = P(X = 1, Y = 0) + P(X = 1, Y = 1) = 0.24 + 0.56 = 0.80 \] \[ P(Y = 0) = P(X = 0, Y = 0) + P(X = 1, Y = 0) = 0.06 + 0.24 = 0.30 \] \[ P(Y = 1) = P(X = 0, Y = 1) + P(X = 1, Y = 1) = 0.14 + 0.56 = 0.70 \] Step 2: Compute the mutual information
Now, we can compute the mutual information: \[ I(X; Y) = P(X = 0, Y = 0) \log_2 \frac{P(X = 0, Y = 0)}{P(X = 0)P(Y = 0)} + \cdots \] After performing the necessary calculations for each term, the mutual information \( I(X; Y) \) simplifies to: \[ I(X; Y) = 0 \] Thus, the correct answer is 0.