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 ai and b are constants
- The mean and variance can be calculated using results from random variables
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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
- 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, ..., Xn
- The sample mean distribution is the distribution of the values of the sample mean
- For an individual sample the sample mean can be calculated
- This is also called a point estimate
- 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
- Therefore you divide the variance of the population by the size of the sample to get the variance of the sample mean distribution
Worked example
Amber makes a cup of tea using a hot drink vending machine. When the hot water button is pressed the machine dispenses ml of hot water and when the milk button is pressed the machine dispenses ml of milk. It is known that and .
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 ml.