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 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
- 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
- An exam question might involve different types of distributions so make it clear which distribution is being used for each variable
The random variable represents the speeds (mph) of a certain species of cheetahs when they run. The variable is modelled using .