The file ProbabilityFunctions.mth defines the incomplete gamma function and its relatives, including various probability densities and cumulative distributions.  Gamma and probability functions are described in Chapters 6 and 26 respectively of Abramowitz and Stegun [1965].  See the introduction to ExponentialIntegrals.mth7RN4D for a discussion of accuracy.  The function definitions in the file are automatically loaded when any of its functions are first used.


POCHHAMMER(a, x) simplifies to (a)x, the Pochhammer's symbol function.  (a)x is defined by the identity (a)x  =  Γ(a+x) / Γ(a).


DIGAMMA_PSI(z) approximates to the digamma function Ψ(z), provided the real part of z is positive (i.e. the phase of z is less than π/2).  Ψ(z) is defined by the identity Ψ(z) = d/dz ln Γ(z) =  d/dz Γ(z) / Γ(z).  The built-in DIGAMMA function is described in Probability FunctionsProbability_Functions.


POLYGAMMA(n, z, m) simplifies to m+1 terms of a series approximation for Ψ(n)(z), the nth polygamma function, provided z is not zero or a negative integer.  Ψ(1)(z), Ψ(2)(z), Ψ(3)(z), Ψ(4)(z) are called the trigamma, tetragamma, pentagamma, and hexagamma functions respectively.  Ψ(n)(z) is defined by the identity Ψ(n)(z) = (d/dz)^n Ψ(z) = (d/dz)^(n+1) ln Γ(z).


INCOMPLETE_GAMMA(z, w) approximates to P(z, w), the incomplete gamma function, provided the real part of z is positive. 


INCOMPLETE_GAMMA_SERIES(z, w, m) simplifies to m+1 terms of a series approximation for P(z, w), the incomplete gamma function.


EULER_BETA(z, w) simplifies to B(z, w), the Euler beta function.  B(z, w) is defined by the identity B(z, w) = Γ(z)·Γ(w) / Γ(z+w).


INCOMPLETE_BETA(x, z, w) simplifies to Bx(z, w), the incomplete beta function.


POISSON_DENSITY(k, t) simplifies to the Poisson probability density defined as exp(-t)·t^k/k! .


POISSON_DISTRIBUTION(k, t) simplifies to the cumulative probability Poisson distribution function defined as the sum of exp(-t)·t^j/j! from j=0 to k.


BINOMIAL_DENSITY(k, n, p) simplifies to the binomial probability density defined as 


                   k        n - k
       COMB(n, k)·p ·(1 - p)     


BINOMIAL_DISTRIBUTION(k, n, p) simplifies to the cumulative probability binomial distribution function defined as


       MIN(k,n)                          
         Σ     BINOMIAL_DENSITY(j, n, p)
         j=0                             


HYPERGEOMETRIC_DENSITY(k, n, m, j) simplifies to the hypergeometric probability density defined as


       COMB(m, k)·COMB(j - m, n - k)
       —————————————————————————————
                 COMB(j, n)


HYPERGEOMETRIC_DISTRIBUTION(k, n, m, j) simplifies to the cumulative probability hypergeometric distribution function defined as


         MIN(k,n,m)                                     
             Σ         HYPERGEOMETRIC_DENSITY(α, n, m, j)
       α=MAX(0,n-j+m)                                   


STUDENT(t, n) approximates to the cumulative probability Student's t-distribution function with n degrees of freedom.  Among other things STUDENT can be used to test with a given confidence level whether a sample comes from a population obeying a Gaussian distribution whose mean is m but whose standard deviation is unknown.  If M denotes the mean and S the standard deviation of a sample of size n, then T=(M-m)/(S/√n) is distributed according to the (non-cumulative) Student's t-distribution with n-1 degrees of freedom.  For example, to determine for a sample size of 20 the value which T will not surpass with a confidence level of 99%, simplify

NSOLVE(STUDENT(t,19) = 99%, t)


F_DISTRIBUTION(F, ν1, ν2) approximates to P(F|ν1, ν2), the cumulative probability variance ratio or F-distribution function with ν1 and ν2 degrees of freedom.


CHI_SQUARE(u, v) approximates to P(χ²|v), the cumulative probability chi-square distribution function, where u=χ².


Other Utility File LibraryG5BS2R 

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