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.
p is the
probability mass function of pX are often used to emphasize that these pertain to a particular random variable |z|<1; the radius of convergence being often larger.
If X = (X1,...,Xd) is a discrete random variable taking values 1</sub>, ..., ''x<sub>d</sub>'')"}},"i":0}}]}'/>(x1, ..., xd) in the X is defined as
X. The power series converges absolutely at least for all complex vectors G(1−)=1, where X is recovered by taking derivatives of X and pX=pY. That is, if E[1]=G(1−)=∑i=0∞p(i)=1. The expectation of kthfactorial moment, E[X!(X−k)!]=G(k)(1−),k≥0. So the variance of k-th raw moment of X is given by GX(t) is the probability generating function (of X).
Functions of independent random variables
editProbability generating functions are particularly useful for dealing with functions of independent random variables. For example:
X_i, i=1,2,\\cdots,N</math> is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and\n<math display=\"block\">S_N = \\sum_{i=1}^N a_i X_i,</math> where the <math>a_i</math> are constant natural numbers, then the probability generating function is given by\n<math display=\"block\">G_{S_N}(z) = \\operatorname{E}(z^{S_N}) = \\operatorname{E} \\left( z^{\\sum_{i=1}^N a_i X_i,} \\right) = G_{X_1}( z^{a_1})G_{X_2}(z^{a_2})\\cdots G_{X_N}(z^{a_N}).</math>\n"},"2":{"wt":" In particular, if <math>X</math> and <math>Y</math> are independent random variables:\n<math display=\"block\">G_{X+Y}(z) = G_X(z) \\cdot G_Y(z)</math> and\n<math display=\"block\">G_{X-Y}(z) = G_X(z) \\cdot G_Y(1/z).</math>\n"},"3":{"wt":" In the above, the number <math>N</math> of independent random variables in the sequence is fixed. Assume <math>N</math> is discrete random variable taking values on the non-negative integers, which is independent of the <math>X_i</math>, and consider the probability generating function <math>G_N</math>. If the <math>X_i</math> are not only independent but also identically distributed with common probability generating function <math>G_X = G_{X_i}</math>, then\n<math display=\"block\">G_{S_N}(z) = G_N(G_X(z)).</math> This can be seen, using the [[law of total expectation]], as follows:\n<math display=\"block\">\n\\begin{align}\nG_{S_N}(z) & = \\operatorname{E}(z^{S_N}) = \\operatorname{E}(z^{\\sum_{i=1}^N X_i}) \\\\[4pt]\n& = \\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)).\n\\end{align}\n</math>\nThis last fact is useful in the study of [[Galton–Watson process]]es and [[compound Poisson process]]es.\n"},"4":{"wt":" When the <math>X_i</math> are not supposed identically distributed (but still independent and independent of <math>N</math>), we have\n<math display=\"block\">G_{S_N}(z) = \\sum_{n \\ge 1} f_n \\prod_{i=1}^n G_{X_i}(z),</math> where <math>f_n = \\Pr(N=n).</math> For identically distributed <math>X_i</math>s, this simplifies to the identity stated before, but the general case is sometimes useful to obtain a decomposition of <math>S_N</math> by means of generating functions.\n"}},"i":0}}]}'>
- If ai are constant natural numbers, then the probability generating function is given by
Y are independent random variables:
N of independent random variables in the sequence is fixed. Assume GN. If the GSN(z)=GN(GX(z)). This can be seen, using the law of total expectation, as follows:
N), we have
Xis, this simplifies to the identity stated before, but the general case is sometimes useful to obtain a decomposition of Pr(X≠c)=0 is p of success in each trial, is p.
{0,1,2⋯}, the number of failures until the G(z)=(p1−(1−p)z)r, which converges for r-fold product of the probability generating function of a
geometric random variable with parameter λ is