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·       Format:

Histogrm(min,max,{pi})

·       Probability density function:

Assuming the pi's are normalized so that

 


\frac{n \sum p_i}{max-min} =1


 

f(x)=p_i

if  x_i \leq x < x_{i+1}

where

x_i=i\frac{max-min}{n}+min

·       Cumulative distribution function:

F(x)=F(x_i)+p_i \frac{x-x_i}{x_{i+1}-x_i}

if  x_i \leq x < x_{i+1}

·       Parameter restrictions:

p_i \geq 0,\displaystyle \sum p_i >0,min \leq max

·       Domain: 

min x max

·       Mean:

\displaystyle\sum_{i=0}^{n} \frac{1}{2}(x_{i+1}+x_1) p_i

·       Mode: 

No unique mode

·       Variance:

\displaystyle\sum_{i=0}^{n} \bigg[ \frac{1}{2}(x_{i+1}+x_1) -\bar x \bigg]^2 p_i

 


 

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