## ABSTRACT

For a discrete random variable X that takes the values x0, x1, x2, …, the mean of X is given by

E X x X x ∞

= ⋅ =∑

(2.1)

where E[X] denotes the mean (expected value or expectation) of X Pr {X = xi} denotes the probability that X takes the value xi (i = 0, 1, 2, …)

If X takes nonnegative integer values only (X = 0, 1, 2, …), then the mean of X is given by

Pr n

E X n X n ∞

= ⋅ =∑

(2.2)

Pr n

X n ∞

= >∑

(2.3)

For a continuous random variable X (−∞ < X < ∞), the mean of X is given by

= ∫E X xf x dx

(2.4)

1 F x dx F x dx ∞

= ⎡ − ⎤ −⎣ ⎦∫ ∫

(2.5)

where E[X] denotes the mean (expected value) of X f(x) is the probability density function of X

and

= ≤ = ∫Pr xF x X x f t dt denotes the cumulative distribution function of X.