4.6.1 The Normal Distribution

The binomial distribution is an example of a discrete probability distribution. The normal distribution is an example of a continuous probability distribution.

What is a continuous random variable?

• A continuous random variable (often abbreviated to CRV) is a random variable that can take any value within a range of infinite values
  • Continuous random variables usually measure something
  • For example, height, weight, time, etc

What is a continuous probability distribution?

• A continuous probability distribution is a probability distribution in which the random variable is continuous
• The probability of being a particular value is always zero
  • for any value k
  • Instead we define the probability density function  for a specific value
    • This is a function that describes the relative likelihood that the random variable would be close to that value
  • We talk about the probability of  being within a certain range
• A continuous probability distribution can be represented by a continuous graph (the values for along the horizontal axis and probability density on the vertical axis)
• The area under the graph between the points and is equal to 
  • The total area under the graph equals 1
• As for any value k, it does not matter if we use strict or weak inequalities
  • for any value k when X is a continuous random variable

What is a normal distribution? 

• A normal distribution is a continuous probability distribution
• The continuous random variable can follow a normal distribution if:
  • The distribution is symmetrical
  • The distribution is bell-shaped
• If follows a normal distribution then it is denoted 
  • μ is the mean
  • σ² is the variance
  • σ is the standard deviation
• If the mean changes then the graph is translated horizontally
• If the variance increases then the graph is widened horizontally and made shorter vertically to maintain the same area
  • A small variance leads to a tall curve with a narrow centre
  • A large variance leads to a short curve with a wide centre

What are the important properties of a normal distribution? 

• The mean is μ
• The variance is σ²
  • If you need the standard deviation remember to square root this
• The normal distribution is symmetrical about  
  • Mean = Median = Mode = μ
• There are the results:
  • Approximately two-thirds (68%) of the data lies within one standard deviation of the mean (μ ± σ)
  • Approximately 95% of the data lies within two standard deviations of the mean (μ ± 2σ)
  • Nearly all of the data (99.7%) lies within three standard deviations of the mean (μ ± 3σ)

What can be modelled using a normal distribution? 

• A lot of real-life continuous variables can be modelled by a normal distribution provided that the population is large enough and that the variable is symmetrical with one mode
• For a normal distribution  can take any real value, however values far from the mean (more than 4 standard deviations away from the mean) have a probability density of practically zero
  • This fact allows us to model variables that are not defined for all real values such as height and weight

What can not be modelled using a normal distribution? 

• Variables which have more than one mode or no mode
  • For example: the number given by a random number generator
• Variables which are not symmetrical
  • For example: how long a human lives for