Encyclopedia>Statistics and Probability>Probability and Combinatorics>Probability Distributions>What Are Probability Distributions?

In statistics, you’ll often work with a random variable $X$. This variable usually represents a set of data from a random trial or a statistical survey. A random variable has several possible outcomes. Which possible outcomes it has depends on the random variable.

Example 1

Let $X$ be the number of dots you get when throwing a six-sided die. That makes $X$ a random variable that denotes the number of dots you can get when throwing the die, which makes 1, 2, 3, 4, 5 and 6 the possible outcomes of $X$.

Example 2

You buy a scratch ticket and scratch it. Let $Y$ be the number of dollars you win. This makes $Y$ a random variable. Most of the time you don’t win anything, so the most regular outcome of $Y$ is 0, but if you win $$10$ the outcome of $Y$ is 10.

In the two examples above you can see that $X$ and $Y$ are very different from each other. The variable $X$ has only six different outcomes, all of which are equally likely to occur. On the other hand, $Y$ has a massive number of possible outcomes, and 0 is much more likely to occur than any other outcome.

Mathematically, we say that $X$ and $Y$ have different probability distributions. A probability distribution is a rule that tells you how likely every outcome is. $P\phantom{\rule{-0.17em}{0ex}}\left(X=k\right)$ means “the probability of $X$ being $k$”.

Theory

In statistics, the term probability distribution is used to describe a specific type of formula. When you read probability distribution, think of it as a formula. Different probability distributions have to satisfy different criteria, just like every other formula you’ve come across in mathematics.

**Note!** It’s very important that you know what each probability distribution applies in each case. By determining this, you’ll know which distribution to use on a given set of data.

Example 3

The probability distribution from Example 1 above is “the probability of 1 is $\frac{1}{6}$, the probability of 2 is $\frac{1}{6}$” and so on. This can be represented in a table like this:

$k$ | 1 | 2 | 3 | 4 | 5 | 6 |

$P\phantom{\rule{-0.17em}{0ex}}\left(X=k\right)$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ |

The most important probability distributions are the binomial distribution, the hypergeometric distribution and the normal distribution.

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Random Variable vs Event