# Physics307L:Schedule/Week 5 agenda/PDF

## From wikipedia

A probability density function is any function f(x) that describes the probability density in terms of the input variable x in a manner described below.

• f(x) is greater than or equal to zero for all values of x
• The total area under the graph is 1:
${\displaystyle \int _{-\infty }^{\infty }\,f(x)\,dx=1.}$

The actual probability can then be calculated by taking the integral of the function f(x) by the integration interval of the input variable x.

For example: the probability of the variable X being within the interval [4.3,7.8] would be

${\displaystyle \Pr(4.3\leq X\leq 7.8)=\int _{4.3}^{7.8}f(x)\,dx.}$

For example, the continuous uniform distribution on the interval [0,1] has probability density f(x) = 1 for 0 ≤ x ≤ 1 and f(x) = 0 elsewhere. The standard normal distribution has probability density

${\displaystyle f(x)={e^{-{x^{2}/2}} \over {\sqrt {2\pi }}}}$

If a random variable X is given and its distribution admits a probability density function f(x), then the expected value of X (if it exists) can be calculated as

${\displaystyle \operatorname {E} (X)=\int _{-\infty }^{\infty }x\,f(x)\,dx}$