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### Enterprise Risk Management Formula Book 4. Statistical distributions

4.1          Probability distribution terminology

Suppose a (continuous) real valued random variable, , has a probability density function (or pdf. Then the probability of  taking a value between  and  where  is infinitesimal, , is .

The expected value of a function  (given this pdf) is defined (if the integral exists) as follows and is also sometimes written :

For  to be a pdf it must exhibit certain basic regularity conditions including .

The mean, variance, standard deviation, cumulative distribution function (cdf or just distribution function), inverse cumulative distribution function (inverse cdf or just inverse function or quantile function), skewness (or skew), (excess) kurtosis, mean excess function, ’th central and non-central moments and entropy are defined as:

The cumulants (sometimes called semi-invariants), , of a distribution, if they exist, are defined via the cumulant generating function, i.e. the power series expansion  of . The mean, standard deviation, skewness and (excess) kurtosis of a distribution are ,  and

The mode of a (continuous) distribution, i.e. , is the value at which  is largest.

The median, upper quartile and lower quartile etc. (or more generally percentile) of a (continuous) distribution are , ,  etc. (or ) respectively.

Definitions of the above for discrete real-valued random variables are similar as long as the integrals involved are replaced with sums and the probability density function by the probability mass function , i.e. the probability of  taking the value .

Some of the above are not well defined or are infinite for some probability distributions.

If a discrete random variable can only take values which are non-negative integers, i.e. from the set  then the probability generating function is defined as:

Characteristic functions and (if they exist) central moments and moment generating functions can nearly always be derived from non-central moments by applying the binomial expansion, e.g. ,  etc. (where  is a constant)

The domain (more fully, the domain of definition or range) of a (continuous) probability distribution is the set of values for which the probability density function is defined. The support of a (discrete) probability distribution is the set of values of  for which  is non-zero. The usual convention for a continuous function is to define the distribution only where the probability density function would be non-zero and for a discrete function (usually) to define the distribution only where the probability mass function is non-zero, in which case the domain/range and support coincide.

The survival function (or reliability function) is the probability that the variable takes a value greater than  (i.e. probability a unit survives beyond time  if  is measuring time) so is:

The hazard function (also known as the failure rate) is the ratio of the pdf to the survival function, so is:

The cumulative hazard function is the integral of the hazard function (i.e. the probability of failure at time  given survival to time , if  is measuring time) so is:

Definitions, characteristics and common interpretations of a variety of (discrete and continuous) probability distributions are given in Appendix A.

The probability that  occurs given that  occurs,  is defined for  as:

For discrete random variables, , , the expected value of  given that  occurs,   is defined as follows, where  is the range of :

The following relationships apply:

If  is a vector of (continuous) random variables then its (multivariate) pdf  and its cdf  satisfy:

The covariance between  and  is  and the (Pearson) correlation coefficient is . The covariance matrix and the (Pearson) correlation matrix for multiple series are the matrices  and  which have as their elements  and  respectively.

4.2          Bayes theorem

Let  be a collection of mutually exclusive and exhaustive events with probability of event  occurring being  for . Then, for any event  such that  the probability, , of  occurring conditional on  occurring (more simply the probability of  given ) satisfies:

A singly conditional probability (i.e. order 1) is e.g. . A doubly conditional probability (i.e. order 2) is e.g. , probability of  occurring given both  and  take specific values. Nil-conditioned conditional probabilities (i.e. order 0) are the marginal probabilities, e.g. . A Bayesian network (more simply Bayesian net) is a directed acyclical graph where each node/vertex, say  is associated with a random variable, say  (often a two-valued, i.e. Boolean, random variable) and with a conditional probability table. For nodes without a parent the table contains just the marginal probabilities for the values that  might take. For nodes with parents it contains all conditional probabilities for the values that  might take given that its parents take specified values.

4.3          Compound distributions

If  are independent identically distributed random variables with moment generating function  and  is an independent non-negative integer-valued random variable then  (with  when ) has the following properties:

For example, the compound Poisson distribution has:  and  where  and

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