Power series derived from a discrete probability distribution
In
probability theory, the probability generating function of a
discrete random variable is a
power series representation (the
generating function) of the
probability mass function of the
random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the
probability mass function for a
random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
If X is a
discrete random variable taking values in the non-negative
integers {0,1, ...}, then the probability generating function of X is defined as
[1]
![{\displaystyle G(z)=\operatorname {E} (z^{X})=\sum _{x=0}^{\infty }p(x)z^{x},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/601d281138657d96ff6c5dd0d2295cfda413f5fd)
where
is the
probability mass function of
. Note that the subscripted notations
and
are often used to emphasize that these pertain to a particular random variable
, and to its
distribution. The power series
converges absolutely at least for all
complex numbers
with
; the radius of convergence being often larger.
If X = (X1,...,Xd) is a discrete random variable taking values in the d-dimensional non-negative
integer lattice {0,1, ...}d, then the probability generating function of X is defined as
![{\displaystyle G(z)=G(z_{1},\ldots ,z_{d})=\operatorname {E} {\bigl (}z_{1}^{X_{1}}\cdots z_{d}^{X_{d}}{\bigr )}=\sum _{x_{1},\ldots ,x_{d}=0}^{\infty }p(x_{1},\ldots ,x_{d})z_{1}^{x_{1}}\cdots z_{d}^{x_{d}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6350e1f6a2701e64ce555cfa61902e903502f137)
where p is the probability mass function of X. The power series converges absolutely at least for all complex vectors
with
Probability generating functions obey all the rules of power series with non-negative coefficients. In particular,
, where
,
x approaching 1 from below, since the probabilities must sum to one. So the
radius of convergence of any probability generating function must be at least 1, by
Abel's theorem for power series with non-negative coefficients.
Probabilities and expectations
The following properties allow the derivation of various basic quantities related to
:
- The probability mass function of
is recovered by taking
derivatives of
,
![{\displaystyle p(k)=\operatorname {Pr} (X=k)={\frac {G^{(k)}(0)}{k!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2040c297af2a9fd7b715befc2bea0a2b02e99019)
- It follows from Property 1 that if random variables
and
have probability-generating functions that are equal,
, then
. That is, if
and
have identical probability-generating functions, then they have identical distributions.
- The normalization of the probability mass function can be expressed in terms of the generating function by
![{\displaystyle \operatorname {E} [1]=G(1^{-})=\sum _{i=0}^{\infty }p(i)=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9ac4537a44a8a5216606c8fcc8c093279b799e)
- The
expectation of
is given by
![{\displaystyle \operatorname {E} [X]=G'(1^{-}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c001dc37d1ea6e1bf6774d7fad40f45a8a6e06)
- More generally, the
factorial moment,
of
is given by
![{\displaystyle \operatorname {E} \left[{\frac {X!}{(X-k)!}}\right]=G^{(k)}(1^{-}),\quad k\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee4b3d97b3bf6a3c0103dc8411147afcbe44528)
- So the
variance of
is given by
![{\displaystyle \operatorname {Var} (X)=G''(1^{-})+G'(1^{-})-\left[G'(1^{-})\right]^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e77ce2135d6d93dab77d15b82cda8c60f61779)
- Finally, the
raw moment of X is given by
![{\displaystyle \operatorname {E} [X^{k}]=\left(z{\frac {\partial }{\partial z}}\right)^{k}G(z){\Big |}_{z=1^{-}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31e75bd0711e82a0772c01f373f1fbdf093fce58)
where X is a random variable,
is the probability generating function (of
) and
is the
moment-generating function (of
).
Functions of independent random variables
Probability generating functions are particularly useful for dealing with functions of
independent random variables. For example:
- If
is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and
![{\displaystyle S_{N}=\sum _{i=1}^{N}a_{i}X_{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7aca6500014ec26fb4b0112a4f55c69f7a52fd73)
- where the
are constant natural numbers, then the probability generating function is given by
.
- In particular, if
and
are independent random variables:
and
.
- In the above, the number
of independent random variables in the sequence is fixed. Let'a assume
is discrete random variable taking values on the non-negative integers, which is independent of the
, and consider it's probability generating function
. If the
are not only independent but also identically distributed with common probability generating function
, then
![{\displaystyle G_{S_{N}}(z)=G_{N}(G_{X}(z)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5de4e9e3de32ca266357b455bfd7522f65f4991)
- This can be seen, using the
law of total expectation, as follows:
![{\displaystyle {\begin{aligned}G_{S_{N}}(z)&=\operatorname {E} (z^{S_{N}})=\operatorname {E} (z^{\sum _{i=1}^{N}X_{i}})\\[4pt]&=\operatorname {E} {\big (}\operatorname {E} (z^{\sum _{i=1}^{N}X_{i}}\mid N){\big )}=\operatorname {E} {\big (}(G_{X}(z))^{N}{\big )}=G_{N}(G_{X}(z)).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c696b17b2376859b07c2c83e0084d7aca523fe3)
- This last fact is useful in the study of
Galton–Watson processes and
compound Poisson processes.
- When the
are not supposed identically distributed (but still independent and independent of
), we have
, where
.
- For identically distributed
s, this simplifies to the identity stated before, but the general case is sometimes useful to obtain a decomposition of
by means of generating functions.
- The probability generating function of an almost surely
constant random variable, i.e. one with
and
is
![{\displaystyle G(z)=z^{c}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/453b8923374760b42112386c10a656a5b7feb303)
- The probability generating function of a
binomial random variable, the number of successes in
trials, with probability
of success in each trial, is
![{\displaystyle G(z)=\left[(1-p)+pz\right]^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b882d226739a271dd2d513336fb42612be9f714f)
- Note: it is the
-fold product of the probability generating function of a
Bernoulli random variable with parameter
.
- So the probability generating function of a
fair coin, is
![{\displaystyle G(z)=1/2+z/2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26ab734631930c54b8de5acc96d7b380251e8631)
- The probability generating function of a
negative binomial random variable on
, the number of failures until the
success with probability of success in each trial
, is
, which converges for
.
- Note that this is the
-fold product of the probability generating function of a
geometric random variable with parameter
on
.
![{\displaystyle G(z)=e^{\lambda (z-1)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be2af52aba39b7d60978d2dcf912f401c39baf6b)
The probability generating function is an example of a
generating function of a sequence: see also
formal power series. It is equivalent to, and sometimes called, the
z-transform of the probability mass function.
Other generating functions of random variables include the
moment-generating function, the
characteristic function and the
cumulant generating function. The probability generating function is also equivalent to the
factorial moment generating function, which as
can also be considered for continuous and other random variables.
- Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley.
ISBN
0-471-54897-9 (Section 1.B9)