In probability, a discrete probability density function of a random variable $X$ is said to be a member of the (a, b, 0) class of distributions if

$frac\{p\_k\}\{p\_\{k-1\}\}\; =\; a\; +\; frac\{b\}\{k\},\; qquad\; k\; =\; 1,\; 2,\; 3,\; dots$

where $p\_k\; =\; P(X\; =\; k)$ (provided $a$ and $b$ exist and are real).

There are only three density functions that satisfy this relationship: the Poisson, binomial and negative binomial distributions.

The a and b parameters

For each density function, the values of $a$ and $b$ can be found using the parameters of the distribution.

$$

begin{array}{l|c|c|c|c} hline & P(X = k) & a & b & p_0 hline mathrm{Poisson}(lambda) & e^{-lambda} , frac{lambda^k}{k!} & 0 & lambda & e^{-lambda} mathrm{Bin}(n,p) & {n choose k} , p^k , (1-p)^{n-k} & - frac{p}{1-p} & (n+1) , frac{p}{1-p} & (1-p)^n mathrm{Neg,Bin}(r, beta) & {k+r-1 choose k} , left(frac{1}{1+beta} right)^r , left(frac{beta}{1+beta} right)^k & frac{beta}{1+beta} & (r-1) , frac{beta}{1+beta} & left(frac{1}{1+beta} right)^r hline end{array}

Plotting

An easy way to quickly determine whether a given sample was taken from a distribution from the (a,b,0) class is by graphing the ratio of two consecutive observed data (multiplied by a constant) against the x-axis.

By multiplying both sides of the recursive formula by $k$, you get

$k\; ,\; frac\{p\_k\}\{p\_\{k-1\}\}\; =\; ak\; +\; b,$

which shows that the left side is obviously a linear function of $k$. When using a sample of $n$ data, an approximation of the $p\_k$'s need to be done. If $n\_k$ represents the number of observations having the value $k$, then $hat\{p\}\_k\; =\; frac\{n\_k\}\{n\}$ is an unbiased estimator of the true $p\_k$.

Therefore, if a linear trend is seen, then it can be assumed that the data is taken from an (a,b,0) distribution. Moreover, the slope of the function would be the parameter $a$, while the ordinate at the origin would be $b$.

References

• Klugman, Stuart; Panjer, Harry; Gordon, Willmot (2004). Loss Models: From Data to Decisions, 2nd edition, New Jersey: Wiley Series in Probability and Statistics. ISBN 0-471-21577-5