Linear combinations of random variables

Question 1

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

Answer 1

Two or more samples are often combined in some way to give a composite sample. The expected values for such samples have predictable properties.

𝐸(𝑎) = 𝑎
The mean value of a constant 𝑎 is 𝑎.
𝐸(𝑎𝑋) = 𝑎 × 𝐸(𝑋)
If each value in a probability distribution is multiplied by 𝑎
the mean of distribution will be multiplied by a factor of 𝑎.
𝐸(𝑋 + 𝑏) = 𝐸(𝑋) + 𝑏
If a constant value 𝑏 is added or subtracted from each value
in probability distribution, the mean of the distribution will be
increased or decreased by 𝑏.
∴ 𝐸(𝑎𝑋 + 𝑏) = 𝑎 × 𝐸(𝑋) + 𝑏

Question 2

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

Answer 2

We have seen how the expected value of 𝑎𝑋 + 𝑏 is related to the expected value of 𝑋 the
result is 𝐸(𝑎𝑋 + 𝑏) = 𝑎 × 𝐸(𝑋) + 𝑏. There is a similar result for the variance of 𝑎𝑋 + 𝑏.

𝑉𝑎𝑟(𝑎) = 0
The variance of a constant 𝑎 is 0.
𝑆𝑑(𝑎) = 0
The standard deviation of a constant 𝑎 is 0.
𝑉𝑎𝑟(𝑎𝑋) = 𝑎² × 𝑉𝑎𝑟(𝑋)
If each value in a probability distribution is multiplied by 𝑎 the variance of distribution will be
multiplied by a factor of𝑎²

.
𝑆𝑑(𝑎𝑋) = |𝑎| × 𝑆𝑑(𝑋)
If each value in a probability distribution is multiplied by 𝑎the standard deviation of distribution
will be multiplied by a factor ofȁ𝑎ȁ.
𝑉𝑎𝑟(𝑋 + 𝑏) = 𝑉𝑎𝑟(𝑋)
If a constant value 𝑏 is added or subtracted from each value in probability distribution, the
variance of the distribution will be unchanged.
𝑆𝑑(𝑋 + 𝑏) = 𝑆𝑑(𝑋)
If a constant value 𝑏 is added or subtracted from each value in probability distribution, the
variance of the distribution will be unchanged.
∴ 𝑉𝑎𝑟(𝑎𝑋 + 𝑏) = 𝑎
2 × 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

If X and Y are 2 independent random variables, the expectation of the sum and difference of X and Y is the sum and difference of the 2 single variance.

If X and Y are 2 independent random variables, the variance of the sum and difference of X and Y is the sum of the 2 single variance.

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

Linear combination of random variables

Answer 4

If X and Y are independent random variables, and Z = aX + bY then:

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

If you have 2 poisson variables,
X~Po(λₓ) and Y~Po(λᵧ) then,
E(X + Y) = E(X) + E(Y) = λₓ + λᵧ

Since with poisson variables the variances are equal to their corresponding means,
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