Enterprise Risk Management Formula Book
Appendix A.1: Probability Distributions:
Discrete (univariate) distributions
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A.1 Discrete (univariate) distributions:
Binomial (and Bernoulli), Poisson
Distribution name

Binomial
distribution

Common notation


Parameters

= number of
(independent) trials, positive integer
= probability
of success in each trial,

Support


Probability mass
function


Cumulative distribution
function


Mean


Variance


Skewness


(Excess) kurtosis


Characteristic function


Other comments

Corresponds to the number of successes in a sequence of independent
experiments each of which has a probability of
being successful.
The Bernoulli distribution is and
corresponds to the likelihood of success of a single experiment. Its
probability mass function and cumulative distribution function are:
The Bernoulli distribution with , i.e. , has the
minimum possible excess kurtosis, i.e. .
The mode of is if is 0 or not
an integer and is if . If then the
distribution is bimodal, with modes and .

Distribution name

Poisson distribution

Common notation


Parameters

= event rate
()

Support


Probability mass
function


Cumulative distribution
function

(can also be expressed using the incomplete gamma
function)

Mean


Variance


Skewness


(Excess) kurtosis


Characteristic function


Other comments

Expresses the probability of a given number of events
occurring in a fixed interval of time if the events occur with a known
average rate and independently of the time since the last event.
The median is approximately .
The mode is if is
not integral. Otherwise the distribution is bimodal with modes and
.

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