This is a discrete distribution of a random variable. An example may tell us what it can be used for:

Suppose this Wiki is visited by 12 people daily (as an average). What is the probability, that the Wiki will have 20 visitors a day? The answer is calculated using the Poisson distribution:

20 -12 12 * e p = ------------ 20!

The answer is: p = 0.0097 which is 0.97%. Another example is: Simulating a server system.

Here is an implementation that draws integer numbers being Poisson distributed. It is based on an ecology modelling book I am currently reading and this thread on c.l.t [L1 ] helped finding a good Tcl implementation. That's a discussion on how to draw random numbers from a given probability distribution (not just the onw shown here).

proc RandomPoisson {lambda count} { # # generate random numbers that are poisson distributed # # lambda -> expected value = "mean value" (which happens to also be the variance) # count -> number of numbers to be generated # # (adapted from: "Parameter Estimation in Ecology" by O. Richter & D. Söndgerath) # factorial f (here: the factorial of 0): set f 1 # poisson probability of the value 0: set su [expr {exp(-$lambda)}] set p(0) $su # probabilities of the integers up to the math limits of Tcl: # (computed as the discrete density function) set i 0 while {1} { incr i set f [expr {$f*$i}] set su [expr {$su + exp(-$lambda) * pow($lambda,$i)/double($f)}] if {$su > 1} { # we cannot calculate more precisely here, # so we assume 1 is ok for the final density: set p($i) 1 break } set p($i) $su } # calculate random values according to the # given discrete probability density function: # (this does work for all dicrete distributions, # not only for poisson) for {set c 0} {$c < $count} {incr c} { # random number in the interval [0,1]: set x [expr {rand()}] # transform this number to the correct target interval: for {set j 0} {$j <= $i} {incr j} { if {$p($j) > $x} { lappend result $j break } } } return $result }

We run into a problem here when using Tcl's normal expr. The numbers in the formula quickly put expr to its limits and therefor I have added the case where the probability is just set to 1, when this point arrives. You could use mpexpr instead if you need a better approximation of the very improbable part of the Poisson distribution.

Here is a test:

# produce 10,000 random integer numbers according to the Poisson distribution: set data [RandomPoisson 3.5 10000] # count their frequencies: foreach el $data { if {[info exists count($el)]} { incr count($el) 1 } else { set count($el) 1 } } # and display their frequencies: foreach el [lsort -integer [array names count]] { puts "$el => $count($el)" }

EKB I just found this page after doing an implementation of my own, based on a book I'm reading on simulation & Monte Carlo techniques. I don't think my code is preferable to this, but just for the sake of variety, I'm pasting it here. It also generates a little test code that shows the pdf overlaying a sample of random deviates, to see how they compare.

Note that this does *not* work well for large values of mu. Faster algorithms are available for larger mu.

# Generate a poisson-distributed random deviate # Use algorithm in section 4.9 of Dagpunar, J.S, # "Simulation and Monte Carlo: With Applications # in Finance and MCMC", pub. 2007 by Wiley proc random-poisson {mu number} { set W0 [expr {exp(-$mu)}] set retval {} for {set i 0} {$i < $number} {incr i} { set W $W0 set R [expr {rand()}] set X 0 while {$R > $W} { set R [expr {$R - $W}] incr X set W [expr {$W * $mu/double($X)}] } lappend retval $X } return $retval } proc pdf-poisson {mu x} { # To avoid dividing large values, build up this way: set W [expr {exp(-$mu)}] for {set z 1} {$z <= int($x)} {incr z} { set W [expr {$W * $mu/double($z)}] } return $W } proc cdf-poission {mu x} { # To avoid dividing large values, build up: set W [expr {exp(-$mu)}] set retval $W for {set z 1} {$z <= int($x)} {incr z} { set W [expr {$W * $mu/double($z)}] set retval [expr {$retval + $W}] } return $retval } ########################################################################################## ## ## TESTING ## ## Can test pdf & cdf by running in a console. For random numbers, generate histograms: ## ########################################################################################## package require math::statistics canvas .c pack .c -side top frame .f pack .f -side bottom label .f.mul -text "mu" entry .f.mue -textvariable mu pack .f.mul -side left pack .f.mue -side left button .f.run -text "Run" -command runtest pack .f.run -side left proc runtest {} { set vals [random-poisson $::mu 5000] set remainder 5000 for {set x 0.0} {$x < 20.0} {set x [expr {$x + 1}]} { lappend bins $x set distval [pdf-poisson $::mu $x] set distval [expr {int(5000 * $distval)}] lappend distcounts $distval } # Assume none are left lappend distcounts 0.0 set bincounts [::math::statistics::histogram $bins $vals] .c delete hist .c delete dist math::statistics::plot-scale .c 0 20 0 [math::statistics::max $bincounts] math::statistics::plot-histogram .c $bincounts $bins hist math::statistics::plot-histogram .c $distcounts $bins dist .c itemconfigure dist -fill {} -outline green } set mu 5 runtest

See also: Gaussian Distribution