27. Let X be a Poisson random variable with parameter . Show that the maximum of P...

27. Let X be a Poisson random variable with parameter λ. Show that the maximum of P (X = i) occurs at [λ], where [λ] is the greatest integer less than or equal to λ. Hint: Let p be the probability mass function of X. Prove that p(i) = λ i p(i − 1). Use this to find the values of i at which p is increasing and the values of i at which it is decreasing.