# Which distribution fits which random variable?

A curve, or a function, or a range of values of a variable, is discrete if it has gaps in it – it jumps from one value to another. In practice in S2 discrete variables are variables which only have whole-number values, like number of heads when you toss a coin, or number of goals in a football season.

It is continuous if it has no gaps in it. In other words, you can draw the curve or the range of values without taking the pen off the paper

A discrete random variable which is like the number of heads got from tossing a coin a fixed number of (discrete) times follows a binomial distribution.

A discrete random variable which is like the number of goals scored by a football team over a certain length of (continuous) time follows a Poisson distribution.

A continuous random variable which is like the length of widgets manufactured to a standard length but with random errors follows a normal distribution.

A continuous random variable which is like the waiting time for a train if you arrive randomly at the station but the train runs exactly every 15 minutes follows a continuous uniform distribution.

Binomial is approximated by Poisson for small p and large n

Binomial is approximated by normal for p around 0.5 and middling n

Binomial for any p with n large enough, and for Poisson with large λ, are also approximated by normal.