Editing Chernoff bound (section)

=== Bounds from below from the MGF ===
Using only the moment generating function, a bound from below on the tail probabilities can be obtained by applying the [[Paley–Zygmund inequality|Paley-Zygmund inequality]] to <math>e^{tX}</math>, yielding: <math display="block">\operatorname P \left(X > a\right) \geq \sup_{t > 0 \and M(t) \geq e^{ta}} \left( 1 - \frac{e^{ta}}{M(t)} \right)^2 \frac{M(t)^2}{M(2t)}</math>(a bound on the left tail is obtained for negative <math>t</math>). Unlike the Chernoff bound however, this result is not exponentially tight.

Theodosopoulos<ref>{{Cite journal |last=Theodosopoulos |first=Ted |date=2007-03-01 |title=A reversion of the Chernoff bound |url=https://www.sciencedirect.com/science/article/pii/S0167715206002884 |journal=Statistics & Probability Letters |language=en |volume=77 |issue=5 |pages=558–565 |doi=10.1016/j.spl.2006.09.003 |s2cid=16139953 |issn=0167-7152|arxiv=math/0501360 }}</ref> constructed a tight(er) MGF-based bound from below using an [[exponential tilting]] procedure.

For particular distributions (such as the [[Binomial distribution|binomial]]) bounds from below of the same exponential order as the Chernoff bound are often available.