pmf with unknown K: mean, variance

i) Value of K

The probabilities of a pmf must add up to 1:

K + 0.1 + 0.2 + 2K + 3K + 0.1 = 1 6K + 0.4 = 1 → 6K = 0.6 → K = 0.1

X      -2    -1    0     1     2     3
P(X)   0.1   0.1   0.2   0.2   0.3   0.1

All values are non-negative and sum to 1.

ii) Mean and variance

E(X) = Σ x P(x) = (−2)(0.1) + (−1)(0.1) + 0(0.2) + 1(0.2) + 2(0.3) + 3(0.1) = −0.2 − 0.1 + 0 + 0.2 + 0.6 + 0.3 = 0.8

E(X²) = Σ x² P(x) = 4(0.1) + 1(0.1) + 0 + 1(0.2) + 4(0.3) + 9(0.1) = 0.4 + 0.1 + 0.2 + 1.2 + 0.9 = 2.8

Variance = E(X²) − [E(X)]² = 2.8 − (0.8)² = 2.8 − 0.64 = 2.16

Answer: K = 0.1, mean = 0.8, variance = 2.16