Editing Probability-generating function (section)

===Probabilities and expectations===

The following properties allow the derivation of various basic quantities related to <math>X</math>:
# The probability mass function of <math>X</math> is recovered by taking [[derivative]]s of <math>G</math>, <math display="block">p(k) = \operatorname{Pr}(X = k) = \frac{G^{(k)}(0)}{k!}.</math>
# It follows from Property 1 that if random variables <math>X</math> and <math>Y</math> have probability-generating functions that are equal, <math>G_X = G_Y</math>, then <math>p_X = p_Y</math>.  That is, if <math>X</math> and <math>Y</math> 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 <math display="block">\operatorname{E}[1] = G(1^-) = \sum_{i=0}^\infty p(i) = 1.</math> The [[expected value|expectation]] of <math>X</math> is given by <math display="block"> \operatorname{E}[X] = G'(1^-).</math> More generally, the <math>k^{th}</math>[[factorial moment]], <math>\operatorname{E}[X(X -  1) \cdots (X - k + 1)]</math> of <math>X</math> is given by <math display="block">\operatorname{E}\left[\frac{X!}{(X-k)!}\right] = G^{(k)}(1^-), \quad k \geq 0.</math> So the [[variance]] of <math>X</math> is given by <math display="block">\operatorname{Var}(X)=G''(1^-) + G'(1^-) - \left [G'(1^-)\right ]^2.</math> Finally, the {{mvar|k}}-th [[raw moment]] of X is given by <math display="block">\operatorname{E}[X^k] = \left(z\frac{\partial}{\partial z}\right)^k G(z) \Big|_{z=1^-}</math>
# <math>G_X(e^t) = M_X(t)</math> where ''X'' is a random variable, <math>G_X(t)</math> is the probability generating function (of <math>X</math>) and <math>M_X(t)</math> is the [[moment-generating function]] (of <math>X</math>).