Physics307L F09: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:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \operatorname{E}(X)=\int_{-\infty}^\infty x\,f(x)\,dx}