Central Limit Theorem

Linear Combination of Normals

Suppose that X and Y are independent normal random variables.

• Let X ~ N(m₁, s₁²) and Y ~ N(m₂, s₂²), then X + Y is a normal random variable with mean m₁ + m₂ and variance s₁² + s₂².

We can go a bit further: if a and b are constants then:

• aX + bY ~ N(am₁ + bm₂ , a²s₁² + b²s₂²)

The Central Limit Theorem

The following is an important result known as the central limit theorem:

If X₁, … X_n is are independent random variables random sample from any distribution which has mean m and variance s², then the distribution of X₁+X₂+…+X_n is approximately normal with mean nm and variance ns².

In particular, the distribution of the sample mean, which is (X₁ + X₂ +…+ X_n)/n, is approximately normal with mean m and variance s²/n (since we have multiplied X₁+X₂+…+X_n by (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.