Poisson distribution probabilities and recursion
relationship

The Poisson distribution is given by the following

P(x,?)=e ^ -? * ?^x! x=0,1,2,3…..j…..

Where ?>0 is a parameter which is the average value
μ in poisson distribution.

a) show that the maximum poisson probability P(x=j,?) occurs
at approximately the average value, that is ?=j if ?>1.

(hint: you can take the first order derivative of the
natural log of poisson probability, P(x=j, ?) with respect to ? and set it
equal to 0

b) show that when ?<1 the poisson probability is a
monotonically decreasing function of j, i.e, P(0, ?)>P(1, ?)>P(2,
?)….P(j, ?)…. And never has a maximum value

(hint: you can use the recursion relationship of the poisson
distribution to prove this statement)

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