Mathematical Expectation Expected Value Discrete Random Variable

Mathematical Expectation (Expected Value) of a Random Variable

In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the product of the values within the range of the discrete random variable and their respective probabilities of occurrence.

If "x" is a value within the range of the discrete random variable "X" assuming the values x₁, x₂, x₃, ... x_n, with respective probabilities of occurrence p₁, p₂, p₃, ...., p_n, where p₁ + p₂ + p₃ + ....+ p_n = 1, then the "Mathematical Expectation" or the "Expected Value" {represented by E(x)} of "x" is given by

E(x)	=	x1 p1 + x2 p2 + ... + xn pn
	=	Σ xi pi (Or) Σ pi xi       [i = 1, 2, 3, ... n]
	=	Σ px

Expectation of x^r

The mathematical expectation of or the expected value of x^r is defined to be equal to the sum of the product of the values (within the range of the discrete random variable) raised to the power "r" and the probabilities of occurrence of the value.

E(xr)	=	x1r p1 + x2r p2 + ... + xnr pn
	=	Σ (xi)r pi (Or) Σ pi xir       [i = 1, 2, 3, ... n]

Therefore, Expectation of x² ⇒ E(x²) = Σ p_i x_i²

Expectation of the sum of two or more variables

The mathematical expectation of or the expected value of the sum of two or more variables is the sum of the expectations of the variables.

E(x + y + ...) = E(x) + E(y) + ...

Expectation of the product of two or more variables

The mathematical expectation of or the expected value of the product of two or more independent variables is the product of the expectations of the variables.

E(x × y × ...) = E(x) × E(y) × ...

Illustration

If "x" represents a value within the range of a random variable "X" and "c" is a constant show that (i) E(c) = c; (ii) E(cx) = c E(x) and (iii) E(x − x) = 0

Sol.	(i)	We know, E (x) = Σ px. Therefore, E (c) = Σ p c = c Σ p = c × 1 = c Σ pc = p1c + p2c + ... pnc = c [p1 + p2 + ... pn] = c Σ p
	(ii)	We know, E (x) = Σ px. Therefore, E (cx) = Σ p cx = c Σ px = c E(x)
	(iii)	We know, E (x) = Σ px. Therefore, E (x − x) = Σ p (x − x) = Σ px − Σ px = E(x) − x Σ p = E(x) − x × 1 = x − x = 0