This section covers central limit theorem and the linear combination of normals.
Linear Combination of Normals
Suppose that X and Y are independent normal random variables.
Let X ~ N(μ1 , σ12) and Y ~ N(μ2 , σ22), and , then X + Y is a normal random variable with mean ( μ1 + μ2 ) and variance (σ12 ) + σ22).
We can go a bit further: if a and b are constants then:
aX + bY ~ N(aμ1 + bμ2 , a2 σ12 +b2σ22)
The Central Limit Theorem
The following is an important result known as the central limit theorem:
If X1 ,…. Xn is are independent random variables random sample from any distribution which has mean μ and variance σ2 , then the distribution of X1 , + X2 +…….. Xn is approximately normal with meanμ and variance nσ2.
In particular, the distribution of the sample mean, which is ( X1 , + X2 +…….. Xn )/ n, is approximately normal with mean μ and variance \(\frac{σ^2}{n}\) (since we have multiplied X1 , + X2 +…….. Xn by \(\frac{1}{n}\) and multiplying by a constant multiplies the mean by that constant and the variance by the constant squared). This important result will be used in constructing confidence intervals.