(i) To find the probability distribution of Y, consider the possible outcomes when two values of X are chosen:

• Y = 0: This occurs when both values are the same. The probabilities are:
  • P(2, 2) = 0.5 × 0.5 = 0.25
  • P(4, 4) = 0.4 × 0.4 = 0.16
  • P(6, 6) = 0.1 × 0.1 = 0.01
  Total probability for Y = 0 is 0.25 + 0.16 + 0.01 = 0.42.
• Y = 2: This occurs when the values are (2, 4), (4, 2), (4, 6), or (6, 4). The probabilities are:
  • P(2, 4) = 0.5 × 0.4 = 0.2
  • P(4, 2) = 0.4 × 0.5 = 0.2
  • P(4, 6) = 0.4 × 0.1 = 0.04
  • P(6, 4) = 0.1 × 0.4 = 0.04
  Total probability for Y = 2 is 0.2 + 0.2 + 0.04 + 0.04 = 0.48.
• Y = 4: This occurs when the values are (2, 6) or (6, 2). The probabilities are:
  • P(2, 6) = 0.5 × 0.1 = 0.05
  • P(6, 2) = 0.1 × 0.5 = 0.05
  Total probability for Y = 4 is 0.05 + 0.05 = 0.1.

The probability distribution table for Y is:

(ii) The expected value of Y is calculated as:

\(E(Y) = (0 imes 0.42) + (2 imes 0.48) + (4 imes 0.1) = 0 + 0.96 + 0.4 = 1.36\)