Mathematical Expectation
The expected value of a discrete random variable is a weighted average of all possible values of the random variable,
where the weights are the probabilities associated with the corresponding values.
For Discrete Random Variable
X
E(X) = xf (x)
where x represents each possible value of the random variable, and f (x) is the probability of that value occurring.
For Continuous Random Variable Z ∞
E(X) = xf (x) dx
eJ
−∞
where f (x) is the probability density function (p.d.f.) of the random variable.
Expected Value of a Function of a Random Variable
in
Consider a random variable X with p.d.f. f (x) and distribution function F (x). If g(.) is a function such that g(X)
is a random variable and E(g(X)) exists, then
erl
For a continuous random variable: Z ∞
E[g(X)] = g(x)f (x) dx
−∞
For a discrete random variable:
J
X
E[g(X)] = g(x)f (x)
va x
Properties of Expectation
Property 1 :Addition Theorem of Expectation
pa
The expected value of the sum of random variables is the sum of their individual expected values.
If X and Y are random variables, then
lor
E(X + Y ) = E(X) + E(Y )
provided all the expectations exist.
Generalization:
a
The mathematical expectation of the sum of n random variables is equal to the sum of their expectations, provided
all the expectations exists. Symbolically, if X1 , X2 , . . . , Xn are random variables, then
Am
E(X1 + X2 + . . . + Xn ) = E(X1 ) + E(X2 ) + . . . + E(Xn )
Property 2 :Multiplication Theorem of Expectation
The expected value of the product of random variables is the product of their individual expected values.
If X and Y are independent random variables, then:
.
E(XY ) = E(X)E(Y )
Dr
Generalization: The mathematical expectation of the product of a number of independent random variables is
equal to the product of their expectations. Symbolically, if X1 , X2 , . . . , Xn are n random variables, then
E(X1 X2 . . . Xn ) = E(X1 )E(X2 ) . . . E(Xn )
Pn Pn
That is E[ i=1 Xi ] = i=1 E(Xi )
1
Property 3: Constant Multiplication:
If X is a random variable and a is constant, then
E(aψ(X)) = aE(ψ(X))
E(ψ(X) + a) = E(ψ(X)) + a
Qn Qn
That is E[ i=1 Xi ] = i=1 E(Xi )
Corollary
i If ψ(X) = X then,
E(a(X)) = aE((X))
eJ
and
E(X + a) = E(X) + a
in
ii If ψ(X) = 1 then,
E(a) = a
erl
Property 4: Linearity Property
If X is a random variable and a and b are constants, then
J
E(aX + b) = aE(X) + b
provided all the expectations exists.
va
Corollary:
pa
• If b=0, then E(aX) = aE(X)
• If a=1 and b = −X = −E(X), then
lor
E(X − X) = E(X) − E(X) = X − X = 0
0.1 Property 5 :Expectation of a Linear combination of Random Variables
a
If X
P1n, X2 , . . . , XnPare n random variables and if a1 , a2 , . . . , an are n constants, then
n
E[ i=1 ai Xi ] = i=1 ai E(Xi ) provided all the expectations exist.
Am
Property 6
If X ≥ 0 then E(X) ≥ 0.
Property 7
.
If X and Y are two random variables such that Y ≤ X, then E(Y ) ≤ E(X) provided all the expectations exist.
Dr
Property 8
|E(X)| ≤ E(|X|) provided all the expectations exist.
Property 9
If µ′r exists, then µ′s for all 1 ≤ s ≤ r.
Property 10
If X and Y are two random variables, then
E(h(X) + k(Y )) = E(h(X)) + E(k(Y ))
