Question

Suppose we have the following probability mass function.

X	0	2	4	6	8
F(x)	0.1	0.3	0.2	0.3	0.1

a) Determine the cumulative distribution function, F(x).

b) Determine the expected value (mean), E(X) = μ.

c) Determine the variance, V(X) = σ^2

Answer 1

a)

X                        F(X)

0                        f(0) = 0.1

2                        f(0) + f(2) = 0.1 + 0.3 = 0.4

4                        f(0) + f(2) + f(4) = 0.1 + 0.3 + 0.2 = 0.6

6                        f(0) + f(2) + f(4) + f(6) = 0.1 + 0.3 + 0.2 + 0.3 = 0.9

8                        f(0) + f(2) + f(4) + f(6) + f(8) = 0.1 + 0.3 + 0.2 + 0.3 + 0.1 = 1

b) E(X) = 0 * 0.1 + 2 * 0.3 + 4 * 0.2 + 6 * 0.3 + 8 * 0.1 = 4

c) E(X²) = 0² * 0.1 + 2² * 0.3 + 4² * 0.2 + 6² * 0.3 + 8² * 0.1 = 21.6

Var(X) = E(X²) - (E(X))² = 21.6 - 4² = 5.6