Editing Parameter (section)

===Probability theory===
[[File:Poisson pmf.svg|thumb|right|These traces all represent Poisson distributions, but with different values for the parameter &lambda;]]In [[probability theory]], one may describe the [[probability distribution|distribution]] of a [[random variable]] as belonging to a ''family'' of [[probability distribution]]s, distinguished from each other by the values of a finite number of ''parameters''. For example, one talks about "a [[Poisson distribution]] with mean value λ".  The function defining the distribution (the [[probability mass function]]) is:
:<math>f(k;\lambda)=\frac{e^{-\lambda} \lambda^k}{k!}.</math>
This example nicely illustrates the distinction between constants, parameters, and variables. ''e'' is [[Euler's number]], a fundamental [[mathematical constant]]. The parameter λ is the [[mean]] number of observations of some phenomenon in question, a property characteristic of the system.  ''k'' is a variable, in this case the number of occurrences of the phenomenon actually observed from a particular sample.  If we want to know the probability of observing ''k''<sub>1</sub> occurrences, we plug it into the function to get <math>f(k_1 ; \lambda)</math>.  Without altering the system, we can take multiple samples, which will have a range of values of ''k'', but the system is always characterized by the same λ.

For instance, suppose we have a [[radioactivity|radioactive]] sample that emits, on average, five particles every ten minutes.  We take measurements of how many particles the sample emits over ten-minute periods.  The measurements exhibit different values of ''k'', and if the sample behaves according to Poisson statistics, then each value of ''k'' will come up in a proportion given by the probability mass function above.  From measurement to measurement, however, λ remains constant at 5.  If we do not alter the system, then the parameter λ is unchanged from measurement to measurement; if, on the other hand, we modulate the system by replacing the sample with a more radioactive one, then the parameter λ would increase.

Another common distribution is the [[normal distribution]], which has as parameters the mean μ and the variance σ².

In these above examples, the distributions of the random variables are completely specified by the type of distribution, i.e. Poisson or normal, and the parameter values, i.e. mean and variance.  In such a case, we have a parameterized distribution.

It is possible to use the sequence of [[moment (mathematics)|moments]] (mean, mean square, ...) or [[cumulant]]s (mean, variance, ...) as parameters for a probability distribution: see [[Statistical parameter]].