Modelling with Distributions (Cambridge (CIE) A Level Maths): Revision Note

Modelling with Distributions

When should I use a binomial distribution?

• A random variable that follows a binomial distribution is a discrete random variable

• A binomial distribution is used when the random variable counts something

  • The number of successful trials

  • The number of members of a sample that satisfy a criterion (satisfying the criteria can be seen as a successful trial)

• There are four conditions that X must fulfil to follow a binomial distribution

  • There is a fixed finite number of trials (n)

  • The trials are independent

  • There are exactly two outcomes of each trial (success or failure)

  • The probability of success (p)  is constant

When should I use a geometric distribution?

• A random variable that follows a geometric distribution is a discrete random variable

• A geometric distribution is used when the random variable counts something

  • The number of trials until a successful trial

• The conditions that X  must fulfil to follow a geometric distribution are exactly the same as for a binomial distribution except there is no fixed number of trials

  • Instead, the trials will continue until the first time a success occurs

When should I use a normal distribution?

• A random variable that follows a normal distribution is a continuous random variable

• A normal distribution is used when the random variable measures something and the distribution is:

  • Symmetrical

  • Bell-shaped

• A normal distribution can be used to model real-life data provided the histogram for this data is roughly symmetrical and bell-shaped

  • If the variable is normally distributed then as more data is collected the outline of the histogram should get smoother and resemble a normal distribution curve

Can the binomial distribution and the normal distribution be used in the same question?

• Some questions might require you to first use the normal distribution to find the probability of success and then use the binomial distribution

  • Remember a discrete distribution is either a binomial or geometric distribution

• The key is to make sure you are very clear about what each parameter/variable represents