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

Revision Note

Author

Dan

Last updated

4 December 2023

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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 a² 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
- $\frac{x}{a}$ can be written as $\frac{1}{a} X$

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

$X$ is a random variable such that $E (X) = 5$ and $Var (X) = 4$ .

Find the value of:

(i)

E (3 X + 5)

(ii)

Var (3 X + 5)

(iii)

Var (2 - X)

2-2-1-ax--b-we-solution-1-2

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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 $X^{~} P o (λ)$ and $Y^{~} P o (μ)$ then $X + Y^{~} P o (λ + μ)$
If X and Y are two independent normal distributions then X + Y is also a normal distribution
- If $X^{~} N (μ_{1}, {σ_{1}}^{2})$ and $Y^{~} N (μ_{2}, {σ_{2}}^{2})$ then $X + Y^{~} N (μ_{1} + μ_{2}, {σ_{1}}^{2} + {σ_{2}}^{2})$

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 aX – bY 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(aμ₁ ± bμ₂, 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

$X$ and $Y$ are independent random variable such that

$E (X) = 5 and Var (X) = 3$

$E (Y) = - 2 and Var (Y) = 4 .$

Find the value of:

(i)

E (2 X + 5 Y)

(ii)

Var (2 X + 5 Y)

(iii)

Var (4 X - Y)

2-2-1-ax--by-we-solution-1

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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 + X₂means two observations of X are taken and added together
2X and X₁ + X₂both have the same expected value of 2E(X)
2X and X₁ + X₂have 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 + X₂could 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+ E₆where
  - 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

Worked example

$X ~ N (10, 4^{2}) and Y ~ N (- 5, 8^{2})$

Find $P (3 X > 2 Y + 50)$

2-2-1-linear-combs-we-solution-1

Examiner Tip

Be careful with negatives!

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Previous:2.1.1 The Poisson DistributionNext:2.3.1 Probability Density Function

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

Revision Note

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

What does the distribution of aX + b look like?

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

What does the distribution of X + Y look like?

What does the distribution of aX + bY look like?

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

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

You've read 0 of your 5 free revision notes this week

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Join the 100,000+ Students that ❤️ Save My Exams

1. Statistical Sampling

1.1 Sampling & Data Collection

1.1.1 Sampling & Data Collection

1.2 Sampling & Estimation

1.2.1 Unbiased Estimates

1.2.2 Distribution of Sample Means

1.2.3 Confidence Intervals

2. Statistical Distributions

2.1 Poisson Distribution

2.1.1 The Poisson Distribution

2.2 Linear Combinations of Random Variables

2.2.1 Linear Combinations of Random Variables

2.3 Continuous Random Variables

2.3.1 Probability Density Function

2.3.2 E(X) & Var(X) (Continuous)

2.3.3 Continuous Uniform Distribution

2.4 Working with Distributions

2.4.1 Choosing Distributions

2.4.2 Approximations of Distributions

3. Hypothesis Testing

3.1 Hypothesis Testing

3.1.1 Hypothesis Testing

3.1.2 Type I & Type II Errors

3.2 Hypothesis Testing (Discrete Distribution)

3.2.1 Binomial Hypothesis Testing

3.2.2 Poisson Hypothesis Testing

3.3 Hypothesis Testing (Normal Distribution)

3.3.1 Normal Hypothesis Testing

Author: Dan

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