 # The Exponential Distribution A heavy-tailed distribution defined for positive real numbers. The exponential distribution is the classic model for waiting times between random events.

# The Chi-Squared Distribution

This is another distribution that models real numbers greater than zero. It also has only one parameter, known as the “degrees of freedom” or df. Mathematically, it corresponds to the square of the Gaussian distribution, but I won’t reproduce the formula here because it’s a bit complicated. The chi-squared distribution is rarely used to model actual data, but it is an important distribution because many widely used statistical tests have chi-squared distributions as their null distributions (e.g., Pearson’s chi- square test for 2 x 2 tables).

# The Poisson Distribution The Poisson is the most widely used distribution defined on natural numbers (0, 1, 2, ...). Another distribution on natural numbers is the geometric distribution, which is the discrete analog of the exponential distribution.

# The Bernoulli Distribution This is the classic “heads” and “tails” distribution for a single toss of a coin. In my notation, we arbitrarily assign heads to be 1 and tails to be 0.

# The Binomial Distribution

This is the Bernoulli distribution when you’ve made n observations: In this formula, I’ve indexed the n observations using i, and I hope it’s

ZH ^—1 n

Xi and n - / Xi are the numbers of “heads” (or 1s) and

i=1 =1

“tails” (or 0s) respectively.