Modelling with Distributions (Edexcel International A Level Maths: Statistics 2)

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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 Poisson distribution?

 

  • A random variable that follows a Poisson distribution is a discrete random variable
  • A Poisson distribution is used when the random variable counts something
    • The number of occurrences of an event in a given interval of time or space
  • There are three conditions that X  must fulfil to follow a Poisson distribution 
    • The mean number of occurrences is known and finite (λ)
    • The events occur at random
    • The events occur singly and independently

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

4-4-1-modelling-with-distributions-diagram-1

Can the binomial or Poisson 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 a discrete distribution
    • Remember a discrete distribution is either a binomial or Poisson distribution
  • 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.

3-4-1-modelling-distributions-we-solution

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.

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Author: Amber

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