Sample Mean Distribution (DP IB Applications & Interpretation (AI)): Revision Note

Author

Dan Finlay

Last updated

4 June 2025

Combinations of Normal Variables

What is a linear combination of normal random variables?

Suppose you have n independent normal random variables $X_{i} ~ N (μ_{i}, σ_{i}^{2})$ for i = 1,2,3, ..., n
A linear combination is of the form $X = a_{1} X_{1} + a_{2} X_{2} + \dots + a_{n} X_{n} + b$ where a_i and b are constants
The mean and variance can be calculated using results from random variables
- $E (X) = a_{1} μ_{1} + a_{2} μ_{2} + \dots + a_{n} μ_{n} + b$
- $Var (X) = a_{1}^{2} σ_{1}^{2} + a_{2}^{2} σ_{2}^{2} + \dots + a_{n}^{2} σ_{n}^{2}$
  - The variables need to be independent for this result to be true
A linear combination of n independent normal random variables is also a normal random variable itself
- $X ~ N (a_{1} μ_{1} + a_{2} μ_{2} + \dots + a_{n} μ_{n} + b, a_{1}^{2} σ_{1}^{2} + a_{2}^{2} σ_{2}^{2} + \dots + a_{n}^{2} σ_{n}^{2})$
- This can be used to find probabilities when combining normal random variables

What is meant by the sample mean distribution?

Suppose you have a population with distribution X and you take a random sample with n observations X₁, X₂, ..., X_n
The sample mean distribution is the distribution of the values of the sample mean
- $X = \frac{X_{1} + X_{2} + \dots + X_{n}}{n}$
For an individual sample the sample mean $\bar{x}$ can be calculated
- This is also called a point estimate
- $X$ is the distribution of the point estimates

What does the sample mean distribution look like when X is normally distributed?

If the population is normally distributed then the sample mean distribution is also normally distributed
$E (\bar{X}) = E (\frac{X_{1} + X_{2} + \dots + X_{n}}{n}) = \frac{E (X_{1}) + E (X_{2}) + \dots + {E (X}_{n})}{n} = \frac{μ + μ + \dots + μ}{n} = \frac{n μ}{n} = μ$
$Var (\bar{X}) = Var (\frac{X_{1} + X_{2} + \dots + X_{n}}{n}) = \frac{Var (X_{1}) + Var (X_{2}) + \dots + {Var (X}_{n})}{n ²} = \frac{σ ² + σ ² + \dots + σ ²}{n ²} = \frac{n σ ²}{n ²} = \frac{σ^{2}}{n}$
Therefore you divide the variance of the population by the size of the sample to get the variance of the sample mean distribution
- $X ~ N (μ, σ^{2}) \Rightarrow \bar{X} ~ N (μ, \frac{σ^{2}}{n})$

Worked Example

Amber makes a cup of tea using a hot drink vending machine. When the hot water button is pressed the machine dispenses $W$ ml of hot water and when the milk button is pressed the machine dispenses $M$ ml of milk. It is known that $W ~ N (100, 15^{2})$ and $M ~ N (10, 2^{2})$ .

To make a cup of tea Amber presses the hot water button three times and the milk button twice. The total amount of liquid in Amber’s cup is modelled by $C$ ml.

a) Write down the distribution of $C$ .

4-9-1-ib-ai-hl-linear-normal-comb-a-we-solution

b) Find the probability that the total amount of liquid in Amber's cup exceeds 360 ml.

4-9-1-ib-ai-hl-linear-normal-comb-b-we-solution

c) Amber makes 15 cups of tea and calculates the mean $\bar{C}$ . Write down the distribution of $\bar{C}$ .

4-9-1-ib-ai-hl-linear-normal-comb-c-we-solution

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