Probability inequality : $\Pr(X≤λ/2)≤4/λ$

Question

Assume that $X$ has a Poisson distribution with parameter $λ$. then prove this inequality $$\Pr\left( X \le \frac \lambda 2 \right) \le \frac 4 λ$$

Answer 1

Use Chebyshev's inequality

$$\mathbb{P}(|X-\mathbb{E}[X]| \geq a) \leq {\mathbb{Var} \left[ X \right] \over a^2} $$

For the Poisson distribution what is $ \mathbb{Var} \left[ X \right] $?

The event $\{ X \leq \frac{\lambda}{2} \} \subset \{ X \leq \frac{\lambda}{2} \} \cup \{X \geq \frac{3\lambda}{2} \} = \{ |X - \lambda| \geq \frac{\lambda }{2}\}$.