X ${\sim}$ Bin(n,p) random variable, and $ε > 0, p ≤ 1/2$ satisfy $ε²pn ≥ 3.$
I need to Prove that $$P(X ≤ (1 − ε)pn) ≥ e^{\large{−9ε²pn}}$$
Pretty sure I should use the stirling approximation but I don't know how.
some help please?
https://mathworld.wolfram.com/StirlingsApproximation.html (26)
p.s.
at the same question you need to prove that $P(X \ge (1 + ε)pn)≥ e^{−9ε²pn}$
but I guess it's similar...