=+2. Consider a Poisson distributed random variable X whose mean is a positive integer. Demonstrate that Pr(X ) 1 2 , Pr(X

=+2. Consider a Poisson distributed random variable X whose mean λ is a positive integer. Demonstrate that Pr(X ≥ λ) ≥

1 2

, Pr(X ≤ λ) ≥

1 2 .

(Hints: For the first inequality, show that Pr(X = λ + k) ≥ Pr(X = λ − k − 1)

when 0 ≤ k ≤ λ − 1. For the second inequality, show that Pr(X ≤ λ) =  ∞

λ

yλ

λ!

e−ydy and argue that the integrand f(y) satisfies f(λ + y) ≥ f(λ − y) for y ∈ [0, λ].)