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

Combinations of normal variables

What is a linear combination of normal random variables?

• Suppose you have n independent normal random variables  for i = 1,2,3, ..., n

• A linear combination is of the form  where a_i and b are constants

• The mean and variance can be calculated using results from random variables

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

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    • 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

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    • 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

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• For an individual sample of n observations, the sample mean  can be calculated

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    • This is also called a point estimate

  •  is the distribution of the point estimates

    • is a random variable

    • is a particular observation of

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

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

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