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.
