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

Combinations of normal variables

What is a linear combination of normal random variables?

• Suppose you have n independent normal random variables $X_i~\mathrm{N}\left({\mu }_i, {\sigma }_i^2\right)$ 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

  • $\mathrm{E}\left(X\right)=a_1{\mu }_1+a_2{\mu }_2+ \dots +a_n{\mu }_n+b$

    • This result is true whether or not the variables are independent

  • $\mathrm{Var}\left(X\right)=a_1^2{\sigma }_1^2+a_2^2{\sigma }_2^2+ \dots +a_n^2{\sigma }_n^2$

    • 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

  •  $X~\mathrm{N}\left(a_1{\mu }_1+a_2{\mu }_2+ \dots +a_n{\mu }_n+b,a_1^2{\sigma }_1^2+a_2^2{\sigma }_2^2+ \dots +a_n^2{\sigma }_n^2\right)$

    • This has the expected mean and variance from above

    • The 'extra bit' is that a linear combination of independent normal random variables is also a normal random variable

  • 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

  • $\overline{X}=\frac{X_1+X_2+ \dots +X_n}{n}$

• For an individual sample of n observations, the sample mean $\overline{x}$ can be calculated

  • $\overline{x}=\frac{x_1+x_2+ \dots +x_n}{n}$

    • This is also called a point estimate

  • $\overline{X}$ is the distribution of the point estimates

    • $\overline{X}$ is a random variable

    • $\overline{x}$ is a particular observation of $\overline{X}$

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

• $\mathrm{E}\left(\overline{X}\right)=\mathrm{E}\left(\frac{X_1+X_2+ \dots +X_n}{n}\right)=\frac{\mathrm{E}\left(X_1\right)+\mathrm{E}\left(X_2\right)+ \dots +{\mathrm{E}(X}_n)}{n}=\frac{\mu +\mu + \dots +\mu }{n}=\frac{n\mu }{n}=\mu$

• $\mathrm{Var}\left(\overline{X}\right)=\mathrm{Var}\left(\frac{X_1+X_2+ \dots +X_n}{n}\right)=\frac{\mathrm{Var}\left(X_1\right)+\mathrm{Var}\left(X_2\right)+ \dots +{\mathrm{Var}(X}_n)}{n²}=\frac{\sigma ²+\sigma ²+ \dots +\sigma ²}{n²}=\frac{n\sigma ²}{n²}=\frac{{\sigma }^2}{n}$

• Therefore you divide the variance of the population by the size of the sample to get the variance of the sample mean distribution

  • $X~\mathrm{N}\left(\mu ,{\sigma }^2\right)\Rightarrow \overline{X}~\mathrm{N}\left(\mu ,\frac{{\sigma }^2}{n}\right)$