AQA A Level Maths: Statistics复习笔记4.4.1 Modelling with Distributions

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 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
  • These questions normally involve some sort of sampling
  • The key is to make sure you are very clear about what each parameter/variable represents

Worked Example

In a population of cows, the masses of the cows can be modelled using a normal distribution with mean 550 kg and standard deviation 80 kg. A farmer classifies cows as beefy if they weigh more than 700 kg. The farmer takes a random sample of 10 cows and weighs them.

Find the probability that at most one cow is beefy.


Exam Tip

  • Always state what your variables and parameters represent.  Make sure you know the conditions for when each distribution is (or is not) a suitable model.