Mean of a Random Variable

Mean is the middle or average value of a random variable.

The mean of a random variable is denote by μ.

μ = ∑_i=1ⁿ x_ip_i

X or Xi	x1	x2	x3	...	xn
P(X) or pi	p1	p2	p3	...	pn

The mean of random variable X is also called the expectation of X, and is denoted by E(X).

E(X) = μ = ∑_i=1ⁿ x_ip_i = x₁p₁ + x₂p₂ + x₃p₃ + .... + x_np_n

Ex:

Consider an experiment of tossing three coins simultaneously.

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

X denotes the number of heads in each outcome.

The possible values of X are 0, 1, 2 and 3.

Probability distribution of X:

X or xi	0	1	2	3
P(X) or pi	1/8	3/8	3/8	1/8

μ = ∑_i=1⁴ x_ip_i = x₁p₁ + x₂p₂ + x₃p₃ + x₄p₄

   = (0 x 1/8) + (1 x 3/8) + (2 x 3/8) + (3 x 1/8)

   = 0 + 3/8+ 6/8+ 3/8

   = 12/8 = 1.5

Hence, mean of random variable X (μ) = 1.5