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

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

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