## What do probability distributions tell us?

A probability distribution can be almost completely described if we know its shape, its center and its variation. Most of Descriptive Statistics (typically the first semester) is about describing the shape, center and variation of our data set.

There are a lot of formulas to learn in your Stats class. Keep in mind that almost all of them in the first semester are used to help us characterize each type of distribution we learn.

Here are some examples:

Binomial Distribution—its shape, center and distribution are defined by:

$p(x)= \displaystyle \binom{n}{x} p^x(1-p)^{n-x}; \; \mu = np; \; \sigma= \sqrt{np(1-p)}$

Geometric Distribution—its shape, center and distribution are defined by:

$p(x)=p(1-p)^{x-1}; \; \mu = \dfrac{1}{p}; \; \sigma = \dfrac{\sqrt{1-p}}{p}$

Exponential Distribution—its shape, center and distribution are defined by:

$p(x)= \dfrac{1}{\beta}e^{-x/\beta}; \; \mu = \beta; \; \sigma = \beta$

Where do these formulas come from? Well, we calculate the mean and standard deviation for a distribution just the way you do for a sample. But the algebra can get a little messy, and unless you are a statistics major, you probably don’t care. Just identify the type of distribution you have, and use the formulas to determine the mean and standard deviation.

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