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# Continuous Uniform Sum Random Variable Generator

The continuous uniform sum distribution is the sum of *k* continuous 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 distribution approaches the normal distribution with a mean of *k*(*a+b*)/2 and a variance of *k*(*b-a*)²/12.

You can use the tool below to generate *N* random variables from a continuous uniform sum distribution.

The mean, median, mode, variance, and skewness of the continuous 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*)²/12

skewness = 0

One way to approximate the standard normal distribution with the uniform sum distribution is to set

*a*=

*b*= 0.5 and

*k*= 12. This gives a symmetric bell-shaped probability curve with a mean of 0 and variance of 1.

For dice simulation and other discretely valued sums, use the

*Discrete*Uniform Sum Distribution Random Variable Generator instead of the calculator above.

If you try to use the tool above to simulate dice by setting min = 1, max = 6, and decimal precision = integer, you will generate a discrete sum distribution in which the tail-end values (

*k*and 6

*k*) are under-weighted. This will not give you an accurate simulation.

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