Cumulative Distribution Functions

A cumulative distribution function is a nondecreasing function defined on the interval $\left(\vphantom{ -\infty ,\infty }\right.$ - ∞,∞$\left.\vphantom{ -\infty ,\infty }\right)$, with values in the interval $\left[\vphantom{ 0,1}\right.$0, 1$\left.\vphantom{ 0,1}\right]$. The definition of a distribution function generally describes only the values where the function is positive, the implicit assumption being that the distribution function is zero up to that point. For discrete cumulative distribution functions, the definition also gives only the values where the function changes, the implicit assumption being that the cumulative distribution function is a step function. Commonly, definitions of these functions are stated only for integers. The definition of a density function also generally describes only the values where the function is positive, the implicit assumption being that the function is zero elsewhere.

These distribution and density functions satisfy the relationships

f (x) = $\displaystyle {\frac{{d}}{{dx}}}$F(x) 6pt  
F(x) = $\displaystyle \int_{{-\infty }}^{{x}}$f (u)du 6pt  

Also note that the cumulative distribution function satisfies $\lim_{{x\rightarrow \infty }}^{}$F(x) = 1 and $\lim_{{x\rightarrow -\infty }}^{}$F(x) = 0. In Scientific Notebook, cumulative distribution functions are named $\limfunc$FunctionDist, and the density functions are named $\limfunc$FunctionDen. For example, the probability density functions for the normal distributions are called $\limfunc$NormalDen.

Scientific Notebook includes several families of distributions: Normal, Cauchy, Student's t, Chi-square, F, Exponential, Weibull, Gamma, Beta, Uniform, Binomial, Poisson, and Hypergeometric.