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• 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 constant

• All 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.