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 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?X?is continuous
• The probability of?X?being a?particular value is always zero
  • P(X = k) = 0?for any value?k
  • Instead we define the?probability density function?f(x)?for a specific 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?X?along the horizontal axis and probability?density?on the vertical axis)
• The?area under the graph?between the points?x = a?and?x = b?is equal to?P(a ≤ X ≤ b)
  • The?total area under the graph equals 1
• As P(X = k) = 0 for any value , it does not matter if we use strict or weak inequalities
  • P(X ≤ k) = P(X < k) for any value?k

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?X 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?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?x = μ
  • Mean = Median = Mode =?μ
• The normal distribution curve has two?points of inflection
  • x =?μ ± ?σ?(one standard deviation away from the mean)
• 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)
• For any value?x?a z-score (or z-value) can be calculated which measures how many standard deviations?x?is away from the mean

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?X?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