Cumulative ascending equations

· Format:

Cumulative (min,max,{x_i},{P_i})

· Probability density function:

$\begin{array}{l}\displaystyle f(x)=\frac{P_{i+1}-P_i}{x_{i+1}-x_i}\end{array}$

$\begin{array}{l}\qquad \text{if} \left.\begin{align} x_i\leq x < x_{i+1} \\ i \in \{0,\dots n+1\} \end{align} \right.\end{array}$

where:

$\begin{array}{l}\displaystyle x_0=min,x_{n+1}=max,P_0=0,P_{n+1}=1\end{array}$

· Cumulative distribution function:

$\begin{array}{l}\displaystyle F(x)=\frac{x-x_i}{x_{i+1}-x_i}(P_{i+1}-P_i)+P_i\end{array}$

$\begin{array}{l}\qquad \text{if} \left.\begin{align} x_i\leq x < x_{i+1} \\ i \in \{0,\dots n+1\} \end{align} \right.\end{array}$

· Parameter restrictions:

$\begin{array}{l}\displaystyle 0 \leq P_i \leq 1,P_i \leq P_{i+1},x_i < x_{i+1},n \geq 0\end{array}$

· Domain:

min ≤ x ≤ max

· Mean:

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

· Mode:

No unique mode

· Variance:

No closed form

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