- A-Level Maths

# Statistical Distributions

A **random variable** is a variable whose value is dependent on the outcome of a random event. This means the value is not known until the experiment is carried out, however the probabilities of all the outcomes can be modelled with a** statistical distribution**.

There are multiple ways of writing out a distribution:

The examples above are all the distribution of a fair, six-sided die. The probability of each outcome is the same, so it is called a **discrete uniform distribution**.

The sum of all probabilities in a distribution must equal 1

## The Binomial Distribution

If you repeat an experiment multiples times (each time is known as a 'trial'), you can model **the number of successful trials with the random variable, X**.

A **binomial distribution** is used when:

There are a fixed number of trials,

*n*There are only two possible outcomes (success or fail)

The probability,

*p*, of success is constantAll trials are independent

If the random variable X is distributed binomially with *n* number of trials and fixed probability of success *p*, it is noted as:

X∼B(n, p)

Generally, you use a calculator to calculate binomial problems, but you can also use the **probability mass function:**

There are two forms of the binomial distribution: **exact **and **cumulative**:

### Exact Binomial Problems

This is when a question asks you to find the probability of there being a specific number of successes out of the number of trials, *n*.