# Linear Combinations of Random Variables(CIE A Level Maths: Probability & Statistics 2)

Author

Dan

Expertise

Maths

## aX+b

#### How are the mean and variance of X related to the mean and variance of aX + b?

• If a and b are constants then the following results are true
• E(aX + b) = aE(X) + b
• Var(aX + b) = a² Var(X)
• Note that the mean is affected by multiplication and addition whereas addition does not change the variance
• The factor of   includes the squared because the values of X are squared in the calculation
• You could try and use the first result and the formula for variance to verify the second result
• Remember a subtraction can be written as an addition
• X – b can be written as X + (-b)
• And division can be written as a multiplication
• can be written as

#### What does the distribution of aX + b look like?

• A linear function is applied to each value of X
• The graphical representation of aX + b is a linear transformation (a translation and a stretch) of the graphical representation of X
• If X follows a normal distribution then aX + b will also follow a normal distribution
• If X ~ N(μ, σ²) then aX + b ~ N(aμ + b, a²σ²)
• If X follows a binomial, geometric or Poisson distribution then aX + b will no longer follow the same type of distribution

#### Worked example

is a random variable such that  and .

Find the value of:

(i)
(ii)
(iii)

## aX + bY

#### How are the means and variances of X and Y related to the mean and variance of X + Y ?

• If X and Y are two random variables then X + Y is the random variable whose values are the sums of each pair containing one value of X and one value of Y
• E(X + Y) = E(X) + E(Y)
• this is true for any random variables X and Y
• Note that E(X Y) = E(X) - E(Y) (see below for more information)
• Var(X + Y) = Var(X) + Var(Y)
• this is true if X and Y are independent
• Note that Var(X - Y) = Var(X) + Var(Y) (see below for more information)

#### What does the distribution of X + Y  look like?

• If X and Y are two independent Poisson distributions then X + Y is also a Poisson distribution
• If and then
• If X and Y are two independent normal distributions then X + Y is also a normal distribution
• If and then

#### What does the distribution of aX + bY look like?

• If X and Y are random variables and a and b are two constants we can combine the results for aX + b and X + Y
• E(aX + bY) = aE(X) + bE(Y)
• this is true for any random variables X and Y
• Var(aX + bY) = a²Var(X) + b²Var(Y)
• this is true if X and Y are independent
• Note that b is squared for the variance so we have
• E(aX - bY) = aE(X) - bE(Y)
• Var(aX - bY) = a²Var(X) + b²Var(Y)
• Notice that the variances of aX + bY and aXbY are the same
• If X and Y are two independent normal distributions then aX + bY is also a normal distribution
• If X ~ N(μ1, σ1²) and Y ~ N(μ2, σ2²) then aX ± bY ~ N(1 ± 2, a²σ1² + b²σ2²)
• Note that aX + bY is no longer Poisson even if X and Y are Poisson
• This holds provided a and b are not 0 or 1

#### Worked example

and  are independent random variable such that

Find the value of:

(i)
(ii)
(iii)

## Linear Combinations

#### For a given random variable X, what is the difference between 2X and X1 + X2?

• 2X means one observation of X is taken and then doubled
• X1 + X2 means two observations of X are taken and added together
• 2X and X1 + X2 both have the same expected value of 2E(X)
• 2X and X1 + X2 have different variances
• Var(2X) = 2²Var(X) = 4Var(X)
• Var(X1 + X2) = 2Var(X)
• Imagine X could take the values 0 and 1
• 2X could then take the values 0 and 2 (2 × 0 = 0 and 2 × 1 = 2)
• X1 + X2 could then take the values 0, 1 and 2 (0 + 0 = 0, 0 +1 = 1, 1 + 1 = 2)
• Sometimes questions may describe the variables in context
• The mass of a carton of half a dozen eggs is the mass of the carton plus the mass of the 6 individual eggs and can be modelled using the random variable
• C + E1 + E2 + E3 + E4 + E5 + E6 where
• C is the mass of a carton
• E is the mass of an egg
• It is not C + 6E because the masses of the 6 eggs could be different

#### How do I use linear combinations of normal random variables to find probabilities?

• If the random variables are normally distributed and independent you might be asked to find probabilities such as
• P(X1 + X2 + X3 > 2Y + 5)
• This could be given in words
• Find the probability that the mass of three chickens (X) is more than 5 kg heavier than double the mass of a turkey (Y)
• To solve these problems:
• STEP 1: Rearrange the inequality to get all the random variables on one side
• P(X1 + X2 + X3 – 2Y > 5)
• STEP 2: Find the mean and variance of the combined normal random variable
• μ = E(X1 + X2 + X3 – 2Y) = E(X1) + E(X2) + E(X3) - 2E(Y)
• σ² = Var(X1 + X2 + X3 – 2Y) = Var(X1) + Var(X2) + Var(X3) + 2² Var(X1)
• STEP 3: Find the required probability using the combined normal distribution
• X1 + X2 + X3 – 2Y ~ N(μ, σ²)
• Use z-values and the table of values

Find

#### Exam Tip

• Be careful with negatives!

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