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# Discrete Uniform Sum Random Variable Generator

The discrete uniform sum distribution is the sum of *k* discrete uniform random variables that are bounded between *a* and *b*. When *k* = 1, the distribution is uniform; when *k* = 2, the distribution is triangular. As *k* grows, the uniform sum model approaches a Gaussian probability model.

You can use the tool below to generate *N* random variables from a discrete uniform sum distribution. For example, if simulating the sum of 3 rolled dice, you set *a* = 1, *b* = 6, and *k* = 3.

The mean, median, mode, variance, and skewness of the discrete uniform sum distribution are:

mean =

*k*(

*a+b*)/2

median =

*k*(

*a+b*)/2

mode =

*k*(

*a+b*)/2 if

*k*is at least 2

variance = [

*k*(

*b - a + 1*)² -

*k*]/12

skewness = 0

The tool above is useful in creating simulations of dice sums. For example, suppose you roll two 8-sided dice (octahedrons) where the faces are labeled with the numbers 1 through 8. If you want to simulate this operation 100 times, select

*a*= 1,

*b*= 8,

*k*= 2, and

*N*= 100. This will generate a set of numbers whose mean, median, and variance are close to 9, 9, and 10.5 respectively.

You can further analyze the simulation using the descriptive statistics calculator.

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