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Table 8 Discrete probability distributions Name Abbreviation Probability function p(x) Range ??X Probability generating function ??X(s) Mean ??X Variance ??X^2 Bernoulli B(1, p) p^x q^{1-x} 0, 1 q + ps p pq Binomial B(n, p) (n choose x) p^x q^{n-x} 0, 1, ..., n (q + ps)^n np npq Discrete uniform 1/n 1, 2, ..., n [s(1-s^n)]/[n(1-s)] (n+1)/2 (n^2-1)/12 Poisson Poisson(??) [e^{-??}??^x]/x! 0, 1, ... e^{-??(1-s)} ?? ?? Geometric (1, 2, ...) G1(p) q^{x-1}p 1, 2, ... ps/(1-qs) 1/p q/p^2 Negative binomial (r, r+1, ...) (x-1 choose r-1) q^{x-r} p^r r, r+1, ... [ps/(1-qs)]^r r/p rq/p^2 Geometric (0, 1, ...) G0(p) p^x q 0, 1, ... q/(1-ps) p/q p/q^2 Negative binomial (0, 1, ...) (r+x-1 choose r-1) p^x q^r 0, 1, ... [q/(1-ps)]^r rp/q rp/q^2 Modified geometric { a/c for x=0, [(ad-bc)/c^2](d/c)^{x-1} for x=1, 2, ... } 0, 1, 2, ... (a-bs)/(c-ds) (ad-bc)/(c-d)^2 [2d(ad-bc)]/(c-d)^3 + ?? - ??^2

          Table 8 Discrete probability distributions
Name Abbreviation Probability function p(x) Range ??X Probability generating function ??X(s) Mean ??X Variance ??X^2
Bernoulli B(1, p) p^x q^{1-x} 0, 1 q + ps p pq
Binomial B(n, p) (n choose x) p^x q^{n-x} 0, 1, ..., n (q + ps)^n np npq
Discrete uniform 1/n 1, 2, ..., n [s(1-s^n)]/[n(1-s)] (n+1)/2 (n^2-1)/12
Poisson Poisson(??) [e^{-??}??^x]/x! 0, 1, ... e^{-??(1-s)} ?? ??
Geometric (1, 2, ...) G1(p) q^{x-1}p 1, 2, ... ps/(1-qs) 1/p q/p^2
Negative binomial (r, r+1, ...) (x-1 choose r-1) q^{x-r} p^r r, r+1, ... [ps/(1-qs)]^r r/p rq/p^2
Geometric (0, 1, ...) G0(p) p^x q 0, 1, ... q/(1-ps) p/q p/q^2
Negative binomial (0, 1, ...) (r+x-1 choose r-1) p^x q^r 0, 1, ... [q/(1-ps)]^r rp/q rp/q^2
Modified geometric { a/c for x=0, [(ad-bc)/c^2](d/c)^{x-1} for x=1, 2, ... } 0, 1, 2, ... (a-bs)/(c-ds) (ad-bc)/(c-d)^2 [2d(ad-bc)]/(c-d)^3 + ?? - ??^2

Table 8 Discrete probability distributions
Name Abbreviation Probability function p(x) Range ??X Probability generating function ??X(s) Mean ??X Variance ??X^2
Bernoulli B(1, p) p^x q^1-x 0, 1 q + ps p pq
Binomial B(n, p) (n choose x) p^x q^n-x 0, 1, ..., n (q + ps)^n np npq
Discrete uniform 1/n 1, 2, ..., n [s(1-s^n)]/[n(1-s)] (n+1)/2 (n^2-1)/12
Poisson Poisson(??) [e^-????^x]/x! 0, 1, ... e^-??(1-s) ?? ??
Geometric (1, 2, ...) G1(p) q^x-1p 1, 2, ... ps/(1-qs) 1/p q/p^2
Negative binomial (r, r+1, ...) (x-1 choose r-1) q^x-r p^r r, r+1, ... [ps/(1-qs)]^r r/p rq/p^2
Geometric (0, 1, ...) G0(p) p^x q 0, 1, ... q/(1-ps) p/q p/q^2
Negative binomial (0, 1, ...) (r+x-1 choose r-1) p^x q^r 0, 1, ... [q/(1-ps)]^r rp/q rp/q^2
Modified geometric a/c for x=0, [(ad-bc)/c^2](d/c)^x-1 for x=1, 2, ... 0, 1, 2, ... (a-bs)/(c-ds) (ad-bc)/(c-d)^2 [2d(ad-bc)]/(c-d)^3 + ?? - ??^2

Added by Dale R.

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