Probability-generating function

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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.

Definition[edit]

Univariate case[edit]

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]

G(z) = \operatorname{E} (z^X) = \sum_{x=0}^{\infty}p(x)z^x,

where p is the probability mass function of X. Note that the subscripted notations GX and pX are often used to emphasize that these pertain to a particular random variable X, and to its distribution. The power series converges absolutely at least for all complex numbers z with |z| ≤ 1; in many examples the radius of convergence is larger.

Multivariate case[edit]

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

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},

where p is the probability mass function of X. The power series converges absolutely at least for all complex vectors z = (z1,...,zd ) ∈ ℂd with max{|z1|,...,|zd |} ≤ 1.

Properties[edit]

Power series[edit]

Probability generating functions obey all the rules of power series with non-negative coefficients. In particular, G(1) = 1, where G(1) = limz→1G(z) 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[edit]

The following properties allow the derivation of various basic quantities related to X:

1. The probability mass function of X is recovered by taking derivatives of G

  p(k) = \operatorname{Pr}(X = k) = \frac{G^{(k)}(0)}{k!}.

2. It follows from Property 1 that if random variables X and Y have probability generating functions that are equal, GX = GY, then pX = pY. That is, if X and Y have identical probability generating functions, then they have identical distributions.

3. The normalization of the probability density function can be expressed in terms of the generating function by

\operatorname{E}(1)=G(1^-)=\sum_{i=0}^\infty f(i)=1.

The expectation of X is given by

 \operatorname{E}\left(X\right) = G'(1^-).

More generally, the kth factorial moment, \textrm{E}(X(X -  1) \cdots (X - k + 1)) of X is given by

\textrm{E}\left(\frac{X!}{(X-k)!}\right) = G^{(k)}(1^-), \quad k \geq 0.

So the variance of X is given by

\operatorname{Var}(X)=G''(1^-) + G'(1^-) - \left [G'(1^-)\right ]^2.

4. G_X(e^{t}) = M_X(t) where X is a random variable, G_X(t) is the probability generating function (of X) and M_X(t) is the moment-generating function (of X) .

Functions of independent random variables[edit]

Probability generating functions are particularly useful for dealing with functions of independent random variables. For example:

  • If X1, X2, ..., Xn is a sequence of independent (and not necessarily identically distributed) random variables, and
S_n = \sum_{i=1}^n a_i X_i,
where the ai are constants, then the probability generating function is given by
G_{S_n}(z) = \operatorname{E}(z^{S_n}) = \operatorname{E}(z^{\sum_{i=1}^n a_i X_i,}) = G_{X_1}(z^{a_1})G_{X_2}(z^{a_2})\cdots G_{X_n}(z^{a_n}).
For example, if
S_n = \sum_{i=1}^n X_i,
then the probability generating function, GSn(z), is given by
G_{S_n}(z) = G_{X_1}(z)G_{X_2}(z)\cdots G_{X_n}(z).
It also follows that the probability generating function of the difference of two independent random variables S = X1X2 is
G_S(z) = G_{X_1}(z)G_{X_2}(1/z).
  • Suppose that N is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function GN. If the X1, X2, ..., XN are independent and identically distributed with common probability generating function GX, then
G_{S_N}(z) = G_N(G_X(z)).
This can be seen, using the law of total expectation, as follows:
 G_{S_N}(z) = \operatorname{E}(z^{S_N}) = \operatorname{E}(z^{\sum_{i=1}^N X_i}) = \operatorname{E}\big(\operatorname{E}(z^{\sum_{i=1}^N X_i}| N) \big) = \operatorname{E}\big( (G_X(z))^N\big) =G_N(G_X(z)).
This last fact is useful in the study of Galton–Watson processes.
  • Suppose again that N is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function GN and probability density f_i = \Pr\{N = i\}. If the X1, X2, ..., XN are independent, but not identically distributed random variables, where G_{X_i} denotes the probability generating function of X_i, then
G_{S_N}(z) = \sum_{i \ge 1} f_i \prod_{k=1}^i G_{X_i}(z).
For identically distributed Xi this simplifies to the identity stated before. The general case is sometimes useful to obtain a decomposition of SN by means of generating functions.

Examples[edit]

G(z) = \left(z^c\right). \,
  • The probability generating function of a binomial random variable, the number of successes in n trials, with probability p of success in each trial, is
G(z) = \left[(1-p) + pz\right]^n. \,
Note that this is the n-fold product of the probability generating function of a Bernoulli random variable with parameter p.
  • The probability generating function of a negative binomial random variable on {0,1,2 ...}, the number of failures until the rth success with probability of success in each trial p, is
G(z) = \left(\frac{p}{1 - (1-p)z}\right)^r.
(Convergence for |z| < \frac{1}{1-p}).
Note that this is the r-fold product of the probability generating function of a geometric random variable with parameter 1−p on {0,1,2 ...}.
G(z) = \textrm{e}^{\lambda(z - 1)}.\;\,


Related concepts[edit]

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.

Notes[edit]

References[edit]

  • Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9 (Section 1.B9)