Question

Simultaneously roll two fair 3-sided dice whose sides are numbered 1, 2, and 3. Let random variable K be the sum of the two dice.

a) Calculate and plot the PMF of K.

b) Determine the expected value of K.

c) Calculate the variance of K.

d) Determine the expected value of K² .

e) Determine the variance of K² .

Answer 1

a) P(K = 2) = P(1,1) = 1/3 * 1/3 = 1/9

P(K = 3) = P(1,2) + P(2,1) = 2 * 1/3 * 1/3 = 2/9

P(K = 4) = P(1,3) + P(2,2) + P(3,1) = 3 * 1/3 * 1/3 = 3/9

P(K = 5) = P(2,3) + P(3,2) = 2 * 1/3 * 1/3 = 2/9

P(K = 6) = P(3,3) = 1/3 * 1/3 = 1/9

b) Expected value of K = E(K) = 2 * 1/9 + 3 * 2/9 + 4 * 3/9 + 5 * 2/9 + 6 * 1/9 = 4

d) Expected value of K² = E(K²) = 2² * 1/9 + 3² * 2/9 + 4² * 3/9 + 5² * 2/9 + 6² * 1/9 = 17.33

c) Variance of K = E(K²) - (E(K))² = 17.33 - 4² = 1.33

e) E(K⁴) = 2⁴ * 1/9 + 3⁴ * 2/9 + 4⁴ * 3/9 + 5⁴ * 2/9 + 6⁴ * 1/9 = 388

Variance of K⁴ = E(K⁴) - E(K²)² = 388 - 17.33² = 87.6711