Linear combinations of random variables

Question 1

Mean of a function of a random variable 𝑬(𝑿) = µ

Answer 1

𝐸(𝑎𝑋 + 𝑏) = 𝑎 × 𝐸(𝑋) + 𝑏

Question 2

Variance of a function of a random variable 𝑽𝒂𝒓(𝑿) = 𝝈² (3)

Answer 2

∴ 𝑉𝑎𝑟(𝑎𝑋 + 𝑏) = 𝑎² × 𝑉𝑎𝑟(𝑋)
∴ 𝑆𝑑(𝑎𝑋 + 𝑏) = |𝑎| × 𝑆𝑑(𝑋)

Note: 𝑉𝑎𝑟(𝑋) = 𝐸(𝑋²) – [E(X)]²

Question 3

Sum and difference of independent random variables

Answer 3

For T = X + Y
E(T) = E(X + Y) = E(X) + E(Y)
Var(T) = Var(X + Y) = Var(X) + Var(Y)

For T = X - Y
E(T) = E(X - Y) = E(X) - E(Y)
Var(T) = Var(X - Y) = Var(X) + Var(Y)

Question 4

Any random variables vs independent random variables

Answer 4

For any random variables:
E(aX + bY) = a x E(x) + b x E(Y)

For independent random variables:
Var(aX + bY) = a² x Var(X) + b² x Var(Y)

Question 5

Linear functions and combinations of normally distributed random variables

Answer 5

Linear combination of independent normal variables are also normally distributed

Question 6

The distribution of the sum of 2 independent Poisson variables

Answer 6

X~Po(λₓ) and Y~Po(λᵧ) then,
E(X + Y) = E(X) + E(Y) = λₓ + λᵧ

Var(X + Y) = Var(X) + Var(Y) = λₓ + λᵧ

Mean and variance of X and Y are equal hence X + Y have a poisson distribution given that X and Y are independent

Note:
Linear combination of independent poisson variables of the for aX + bY can not have a poisson distribution