Derivation Of The Poisson Distribution

Conditional Poisson distribution restricted to positive integers

In probability theory, the nil-truncated Poisson (ZTP) distribution is a sure discrete probability distribution whose back up is the gear up of positive integers. This distribution is also known as the provisional Poisson distribution ^[1] or the positive Poisson distribution.^[two] It is the conditional probability distribution of a Poisson-distributed random variable, given that the value of the random variable is not zero. Thus information technology is impossible for a ZTP random variable to exist zero. Consider for example the random variable of the number of items in a shopper's basket at a supermarket checkout line. Presumably a shopper does not stand up in line with nix to buy (i.e., the minimum purchase is 1 particular), so this miracle may follow a ZTP distribution.^[3]

Since the ZTP is a truncated distribution with the truncation stipulated equally $k > 0$ , one can derive the probability mass function $g (k; λ)$ from a standard Poisson distribution $f (k; λ$ ) equally follows: ^[4]

g(grand;\lambda )=P(Ten=k\mid X>0)={\frac {f(chiliad;\lambda )}{1-f(0;\lambda )}}={\frac {\lambda ^{k}due east^{-\lambda }}{one thousand!\left(1-east^{-\lambda }\right)}}={\frac {\lambda ^{thou}}{(e^{\lambda }-1)k!}}

The mean is

\operatorname {E} [Ten]={\frac {\lambda }{1-e^{-\lambda }}}={\frac {\lambda e^{\lambda }}{due east^{\lambda }-1}}

and the variance is

\operatorname {Var} [10]={\frac {\lambda +\lambda ^{two}}{i-e^{-\lambda }}}-{\frac {\lambda ^{two}}{(i-e^{-\lambda })^{two}}}=\operatorname {East} [X](1+\lambda -\operatorname {Eastward} [Ten])

Parameter interpretation [edit]

The method of moments estimator ${\widehat {\lambda }}$ for the parameter $\lambda$ is obtained past solving

{\frac {\widehat {\lambda }}{1-e^{-{\widehat {\lambda }}}}}={\bar {x}}

where ${\bar {x}}$ is the sample hateful.^[1]

This equation does non accept a closed-form solution. In practice, a solution may be institute using numerical methods.

Generating zip-truncated Poisson-distributed random variables [edit]

Random variables sampled from the Zippo-truncated Poisson distribution may exist achieved using algorithms derived from Poisson distributing sampling algorithms.^[5]

          init:          Allow          g ← 1, t ←          e          ^−λ          / (i -          e          ^−λ) * λ, southward ← t.          Generate uniform random number u in [0,1].          while          due south < u          do:          k ← grand + 1.          t ← t * λ / grand.          s ← south + t.          return          k.

The toll of the procedure in a higher place is linear in k, which may be large for large values of $\lambda$ . Given admission to an efficient sampler for non-truncated Poisson random variates, a non-iterative approach involves sampling from a truncated exponential distribution representing the fourth dimension of the first event in a Poisson indicate process, conditional on such an effect existing.^[vi] A simple NumPy implementation is:

          def sample_zero_truncated_poisson(charge per unit):         u = np.random.uniform(np.exp(-rate), i)         t = -np.log(u)         return one + np.random.poisson(rate - t)

References [edit]

^ ^a ^b Cohen, A. Clifford (1960). "Estimating parameters in a provisional Poisson distribution". Biometrics. sixteen (two): 203–211. doi:10.2307/2527552. JSTOR 2527552.
^ Singh, Jagbir (1978). "A characterization of positive Poisson distribution and its application". SIAM Journal on Practical Mathematics. 34: 545–548. doi:10.1137/0134043.
^ "Stata Data Analysis Examples: Nix-Truncated Poisson Regression". UCLA Institute for Digital Research and Educational activity. Retrieved 7 August 2013.
^ Johnson, Norman L.; Kemp, Adrianne W.; Kotz, Samuel (2005). Univariate Discrete Distributions (third ed.). Hoboken, NJ: Wiley-Interscience.
^ Borje, Gio (2016-06-01). "Goose egg-Truncated Poisson Distribution Sampling Algorithm". Archived from the original on 2018-08-26.
^ Hardie, Ted (1 May 2005). "[R] simulate aught-truncated Poisson distribution". r-aid (Mailing list). Retrieved 27 May 2022.