Question

Exercise 4.30. Show that the probability mass function of a Poisson(x) random variable is first increasing and then decreasing, similarly to the binomial distri- bution. For what range of values is the p.m.f. increasing

Answer 1

The answer to this question can be defined as follows:

Step-by-step explanation:

`\ Given \ values:\\\\\ let \ x \ is \ a \ poisson \ (x). Then,\\\\\ The \ probability \ mass \ function \ of \ x \ is,\\\\p(x= x) = c^(-\lambda) (\lambda^n)/(x!), \\\\x = 0, 1, 2\\ \ consider \ the \ ratios \\\\(p(x=i))/(p(x=i-1)) = (e^(-\lambda)(\lambda!)/(i!))/(e^(-\lambda) ( \lambda^(i-1))/((i-1)!)) = (\lambda)/(i) \\\\`

`\If (\lambda)/(i) > 1 = (p(x-i))/(p(x=i-1)) >1 \\\\p(x=i)>p(x=i-1)\\\\\ Therefore,\ the \ probability \ mass \ method \ x\ increased \ unit \ by \ \ (\lambda)/(i) > 1\\\\\lambda > i\\\\\ If \ \ (\lambda)/(i)<1 \ = (p(x-i))/(p(x=i-1)) <1 \\\\p(x=i)<p(x=i-1)`

`\ The \ probability \ mass \ method \ x \ decreased \ by \ \ (\lambda)/(i)<1\\\\\lambda < i \\\\ \ The \ highest \ value \ counters \ when \\\\ i=[\lambda]\\\\\ If \ \lambda \ is \ number \ then \ the \ highest \ value \ occured \ by \\\\ \ i= \lambda -1 \ and\ i=\lambda`