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:
 * $$ \int_{-\infty}^\infty \,f(x)\,dx = 1. $$

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


 * $$\Pr(4.3 \leq X \leq 7.8) = \int_{4.3}^{7.8} f(x)\,dx.$$

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


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

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


 * $$\operatorname{E}(X)=\int_{-\infty}^\infty x\,f(x)\,dx$$

Links
[http://en.wikipedia.org/wiki/Probability_density_function wikipedia prob. dens. func. article]

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