#### 52.3.3 Poisson Random Variable

The Poisson distribution is a discrete probability distribution. It is the probability that a given number of events occur in a fixed interval when the events occur independently of the time of the last event, and the events occur with a known constant rate.

Function: pdf_poisson (x,m)

  Returns the value at x of the probability function of a \({\it Poisson}(m)\) random variable, with m>0. To make use of this function, write first `load("distrib")`.

  The pdf is $$ f(x; m) = {m^x e^{-m}\over x!} $$

  Categories:Package distrib ·

Function: cdf_poisson (x,m)

  Returns the value at x of the distribution function of a \({\it Poisson}(m)\) random variable, with m>0.

  The cdf is $$ F(x; m) = Q(\lfloor x \rfloor + 1, m) $$

  where Q(x,m) is the gamma_incomplete_regularized function.

      (%i1) load ("distrib")$

      (%i2) cdf_poisson(3,5);
      (%o2)          gamma_incomplete_regularized(4, 5)

      (%i3) float(%);
      (%o3)                  0.2650259152973619

  Categories:Package distrib ·

Function: quantile_poisson (q,m)

  Returns the q-quantile of a \({\it Poisson}(m)\) random variable, with m>0; in other words, this is the inverse of `cdf_poisson`. Argument q must be an element of [0,1]. To make use of this function, write first `load("distrib")`.

  Categories:Package distrib ·

Function: mean_poisson (m)

  Returns the mean of a \({\it Poisson}(m)\) random variable, with m>0. To make use of this function, write first `load("distrib")`.

  The mean is $$ E[X] = m $$

  Categories:Package distrib ·

Function: var_poisson (m)

  Returns the variance of a \({\it Poisson}(m)\) random variable, with m>0. To make use of this function, write first `load("distrib")`.

  The variance is $$ V[X] = m $$

  Categories:Package distrib ·

Function: std_poisson (m)

  Returns the standard deviation of a \({\it Poisson}(m)\) random variable, with m>0. To make use of this function, write first `load("distrib")`.

  The standard deviation is $$ V[X] = \sqrt{m} $$

  Categories:Package distrib ·

Function: skewness_poisson (m)

  Returns the skewness coefficient of a \({\it Poisson}(m)\) random variable, with m>0. To make use of this function, write first `load("distrib")`.

  The skewness is $$ SK[X] = {1\over \sqrt{m}} $$

  Categories:Package distrib ·

Function: kurtosis_poisson (m)

  Returns the kurtosis coefficient of a Poisson random variable Poi(m), with m>0. To make use of this function, write first `load("distrib")`.

  The kurtosis coefficient is $$ KU[X] = {1\over m} $$

  Categories:Package distrib ·

Function: random_poisson (m)
`random_poisson` (m,n)

  Returns a \({\it Poisson}(m)\) random variate, with m>0. Calling `random_poisson` with a second argument n, a random sample of size n will be simulated.

  The implemented algorithm is the one described in Ahrens, J.H. and Dieter, U. (1982) Computer Generation of Poisson Deviates From Modified Normal Distributions. ACM Trans. Math. Software, 8, 2, June,163-179.

  To make use of this function, write first `load("distrib")`.

  Categories:Package distrib ·Random numbers ·