14A - The normal distribution
One of the most important continuous distributions is the Normal distribution which uses a symmetric bell curve as its probability density function: \[f(x)=\frac{1}{\sigma \sqrt{2\pi}}e^{\frac{-1}{2}\left ( \frac{x-\mu}{\sigma} \right )^2}, -\infty<x<\infty\] where \(\mu\) is the mean (or expected value, \(E(X)\)) and \(\sigma\) is the standard deviation for the distribution.
The following summarises key characteristics of the normal distribution:
- The normal curve approaches \(y=0\) as \(x \to \pm \infty\).
- The bell curve is symmetric about \(x=\mu\); therefore, the mean, median and mode are all \(\mu\).
- The area under the normal curve is equal to 1 making it a probability density function.
- Approximately 95% exists within \(2\sigma\) of \(\mu\).
\(Pr(\mu-\sigma<X<\mu+\sigma)=0.68\)
|
\(Pr(\mu-2\sigma<X<\mu+2\sigma)=0.95\)
|
\(Pr(\mu-3\sigma<X<\mu+3\sigma)=0.997\)
|
■ If the random variable is normally distributed then \(X\)~\(N(\mu,\sigma^2)\), where \(\mu\) is the mean and \(\sigma^2\) is the variance.
14A - Example 1: The normal distribution (VCAA, 2003)
[VCAA, 2003 Exam 1 Part 1 Question 23]
The diagram below shows the graphs of two normal distribution curves with means \(\mu_1\) and \(\mu_2\) and standard deviations \(\sigma_1\) and \(\sigma_2\) respectively. Which of the following statements is true?
A. \(\mu_1>\mu_2\) and \(\sigma_1=\sigma_2\) B. \(\mu_1>\mu_2\) and \(\sigma_1>\sigma_2\) C. \(\mu_1=\mu_2\) and \(\sigma_1>\sigma_2\) D. \(\mu_1=\mu_2\) and \(\sigma_1<\sigma_2\) E. \(\mu_1<\mu_2\) and \(\sigma_1=\sigma_2\) |
14A - Example 1: Video solution
Video solution coming soon! Option D is correct. 14A - Example 1: Practice
Question 1: ABC 14A - Example 1: Solutions
Question 1: ABC |