Physics307L F07: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:
- [math]\displaystyle{ \int_{-\infty}^\infty \,f(x)\,dx = 1. }[/math]
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
- [math]\displaystyle{ \Pr(4.3 \leq X \leq 7.8) = \int_{4.3}^{7.8} f(x)\,dx. }[/math]
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
- [math]\displaystyle{ f(x)={e^{-{x^2/2}}\over \sqrt{2\pi}} }[/math]
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
- [math]\displaystyle{ \operatorname{E}(X)=\int_{-\infty}^\infty x\,f(x)\,dx }[/math]
