Discuss the below:
Q: The Poisson distribution is given by the following
P(x,λ)=e ^ -λ * λ^x! x=0,1,2,3.....j.....
Where λ>0 is a parameter which is the average value μ in poisson distribution.
a) Show that the maximum poisson probability P(x=j,λ) occurs at approximately the average value, that is λ=j if λ>1.
(hint: you can take the first order derivative of the natural log of poisson probability, P(x=j, λ) with respect to λ and set it equal to 0
b) Show that when λ<1 the poisson probability is a monotonically decreasing function of j, i.e, P(0, λ)>P(1, λ)>P(2, λ)....P(j, λ).... And never has a maximum value
(hint: you can use the recursion relationship of the poisson distribution to prove this statement)