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

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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

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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").

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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 $$ E[X] = m $$

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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 $$ V[X] = m $$

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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 $$ V[X] = \sqrt{m} $$

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

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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 $$ KU[X] = {1\over m} $$

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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").

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