I'm trying to prove that $e^{-k-1} \leq \frac{1}{k^2}$ for $k > 1$ but I feel I'm missing something (maybe an standard inequality?) Could anyone give me a pointer like "Use such fact"?
Context:
For $k > 1$ and the random variable $X \sim \mbox{Exponential}(\beta)$:
$P(|X-\mu_X| \geq k \sigma_X) = e^{-k-1}$
And using Chebyshev's inequality:
$P(|X-\mu_X| \geq k \sigma_X) \leq \frac{1}{k^2}$
Thanks.
(This is my first question in the site, so please let me know if I did something wrong.)