Comments on What is the Name of Function for Probability of a Certain Sum on Random Die Rolls?
Parent
What is the Name of Function for Probability of a Certain Sum on Random Die Rolls?
Hi.
I'm writing a book about using statistics for roleplaying game design and am using this equation for calculating the probability of rolling a particular sum "n" on "z" throws of an "a"-sided die
What is the name for this function? I'm trying to search for calculators to recommend that support it, and I can't think of the words to search for.
Post
I don't know some special name for exactly that sum, but it is closely related to what are known as polynomial coefficients or extended binomial coefficients. These are often written as
Polynomial coefficients and distribution of the sum of discrete uniform variables by Caiado, C.C.S. and Rathie, P.N. provides this definition and a recurrence relation to compute this number. As the title suggests, it also provides a direct solution to your probability problem. If
The aforementioned paper gives a summation expression in terms of gamma functions which probably does reduce to your sum, but a clearer expression is in Restricted Weighted Integer Compositions and Extended Binomial Coefficients by Steffen Eger. Table 1 in the conclusion (page 22) list various binomial identities and the corresponding extended binomial identities derived in that paper. (See also equation 15 in Example 33.) One of which is:
Matching your notation and performing the adjustment as above gives:
Just for completeness, for the second binomial coefficient factor to be non-zero we need
0 comment threads