AMS 315, Spring 2012

The Standard Normal Distribution

P(x) = 1√2πe-½x²

Z & T

The most important thing in the class is the Confidence Interval.

The Confidence Interval always has the form MEAN ± CV * SE.

MEAN is the sample mean.

CV is the Critical Value, like Zₐ.

SE is the Standard Error, the Standard Deviation of the variable of interest. Often related to the StanDev of another variable, like SE = σ√n

Χ+Y

The second most important thing in the class is adding Normal Distributions.

If I take a sample from Normal population X and another from Normal population Y then add those values, those sums would follow a normal distribution.

If X~N and Y~N then X+Y~N. Specifically, If X~N(0,1) and Y~N(0,1) then X+Y~N(0,2).

Χ²

The third most important thing in the class is squaring Normal Distributions: Chi-Squared.

If I take a sample from Normal population X then square that value, those squares wouldn't follow a Normal Distribution. But a distribution we'll call Χ².

If X~N then X*X≁N, instead X*X~Χ². Specifically, If X~N(0,1) then X*X~Χ².

The Chi-Squared Distribution

If X~N, P(x) = 1√2πe-½x². To find P(Χ²) we do a change of variable. Let Y=x².

PY(Y) = |dxdy| PX(X) = 12√y * 1√2πe-½x² = 12√2πye-½y

Since the Normal Distribution, and Squaring are symmetric we must multiply by 2.

P(Y) = 1√2πye-½y