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·       Format:

Cumulative (min,max,{xi},{Pi})

·       Probability density function:

f(x)=\frac{P_{i+1}-P_i}{x_{i+1}-x_i}
\begin{equation} \qquad \text{if} \left.\begin{align} x_i\leq x < x_{i+1} \\ i \in \{0,\dots n+1\} \end{align} \right. \end{equation}

where: 

x_0=min,x_{n+1}=max,P_0=0,P_{n+1}=1

·       Cumulative distribution function:

F(x)=\frac{x-x_i}{x_{i+1}-x_i}(P_{i+1}-P_i)+P_i
\begin{equation}\qquad \text{if} \left.\begin{align} x_i\leq x < x_{i+1} \\ i \in \{0,\dots n+1\} \end{align} \right. \end{equation}

·       Parameter restrictions:

0 \leq P_i \leq 1,P_i \leq P_{i+1},x_i <x_{i+1},n \geq 0

·       Domain: 

min x max

·       Mean:

\displaystyle\sum_{i=0}^{n} \frac{f(x_i)}{2} \big( x_{i+1}^2-x_i^2\big)

·       Mode: 

No unique mode

·       Variance:

No closed form

 


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