8.8 | Probability, Mean, and Median


A median of a quantity \(x\) distributed through a population is a value \(T\) such that half the population has values of \(x\) less than (or equal to) \(T\), and half the population has values of \(x\) greater than (or equal to) \(T\). Thus, a median satisfies
$$
\int_{-\infty}^T p(x) \;dx = 0.5
$$
where \(p\) is the density function. In other words, half the area under the graph of \(p(x)\) lies to the left of \(x=T\).
If a quantity has density function \(p(x)\), then the mean value of the quantity is given by
$$
\int_{-\infty}^\infty p(x) \; dx.
$$
A normal distribution has a density function of the form
$$
p(x) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-(x- \mu)^2/(2 \sigma^2)}
$$
where \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation, with \(\sigma >0\).

E 8.8 Exercises