A bound related to Poisson random variables

Question

Suppose that $N_{2k}$ is a Poisson random variable with mean $2k$, Exercise 20.6 in the book "Markov Chains and Mixing Times (2nd edition)" written by David A.Levin and Yuval Peres asks us to prove that $$\mathbb{P}(N_{2k} < k) \leq \left(\frac{e}{4}\right)^k.$$ By an easy application of Chebyshev's inequality, I have no issue to show that $$\mathbb{P}(N_{2k} < k) \leq \frac{2}{k}.$$ But I am really stuck on getting the desired inequality. Thanks for any help!

Answer 1

I don't think the result as stated is correct, e.g. for $k=100$:

> ppois(99, 200)
[1] 1.843894e-15
> (exp(1)/4)^100
[1] 1.672819e-17

However, you can get another bound with exponential decay in $k$ using the Chernoff type bound described here:

$$ P(N_{2k} \leq k) \leq \frac{(e2k)^ke^{-2k}}{k^k} = \left(\frac{e^{-2}e2k}{k}\right)^k = \left(\frac{2}{e} \right)^k $$