IB DP Maths: AI HL复习笔记4.8.1 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?

What is a normal distribution? 

• A normal distribution is a continuous probability distribution
• • 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 changes then the graph is stretched horizontally
  • 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
  • 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