- Don’t try to memorize the formulas for all the probability distributions! Instead, it’s much more useful to learn,
- The relationships between distributions
- The basic shapes of the distributions (unimodal? nonnegative? …)
- How their shapes change then their parameters are changed We’ll give a refresher of some common probability distributions in these notes.
Chi-Square Distribution. This distribution arises as the sum of squares of standard normals. That is, if \(z_[k] \sim \mathcal{N}\left(0, 1\right)\), then \(\sum_{k} z_{k}^2 \sim \chi^2_{K}\), a chi-square distribution with \(K\)-degrees of freedom (d.f.).
This distributions claim to fame is that if \(y_i \sim \mathcal{N}\left(\mu, \sigma^2\right)\) independently, then
\[ \frac{1}{\sigma^2}\sum_{i = 1}^{n} \left(y_i- \bar{y}\right)^2 \sim \chi^2_{n -1} \] which is a nontrivial but very useful fact, since the expression on the right is similar to the usual estimator for the sample standard deviation. We’ll make use of connection when we construct some common hypothesis tests.
- \(t\) distribution. A \(t\) distribution with \(k\) d.f. arises as a ratio between a normal and the square root of a chi-square with K d.f.,
\[ \frac{\mathcal{N}\left(0, 1\right)}{\sqrt{\frac{\chi^2_{K}}{K}}} \]
This seems like an esoteric fact, but notice that the usual way of standardizing the mean (when the true variance is unknown) has this form,
\[ \frac{\sqrt{n}\left(\bar{y} - \mu\right)}{S} \]
- \(F\) Distribution. The \(F\) distribution occurs as the ratio of independent chi-squares (suitably rescaled),
\[ F_{u, v} = \frac{\frac{1}{u}\chi^2_u}{\frac{1}{v}\chi^2_v} \]
Since chi-squares arise whenever we have sums of squares, this distribution will come in handy whenever we need to compare two different sums of squares.
library(EBImage)
display(readImage("https://uwmadison.box.com/shared/static/dv5tvok0m9vkqqmkd3c0woam5is7gzse.png"))
