1.5.1 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

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

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