Editing Moment-generating function (section)

==Other properties==
[[Jensen's inequality]] provides a simple lower bound on the moment-generating function:
<math display="block"> M_X(t) \geq e^{\mu t}, </math>
where <math>\mu</math> is the mean of {{mvar|X}}.

The moment-generating function can be used in conjunction with [[Markov's inequality]] to bound the upper tail of a real random variable {{mvar|X}}. This statement is also called the [[Chernoff bound]]. Since <math>x \mapsto e^{xt}</math> is monotonically increasing for <math>t>0</math>, we have
<math display="block"> \Pr(X \ge a) = \Pr(e^{tX} \ge e^{ta}) \le e^{-at} \operatorname{E}\left[e^{tX}\right] = e^{-at}M_X(t)</math>
for any <math>t>0</math> and any {{mvar|a}}, provided <math>M_X(t)</math> exists. For example, when {{mvar|X}} is a standard normal distribution and <math>a > 0</math>, we can choose <math>t=a</math> and recall that <math>M_X(t)=e^{t^2/2}</math>. This gives <math>\Pr(X\ge a)\le e^{-a^2/2}</math>, which is within a factor of {{math|1+''a''}} of the exact value.

Various lemmas, such as [[Hoeffding's lemma]] or [[Bennett's inequality]] provide bounds on the moment-generating function in the case of a zero-mean, bounded random variable.

When <math>X</math> is non-negative, the moment generating function gives a simple, useful bound on the moments:
<math display="block">\operatorname{E}[X^m] \le \left(\frac{m}{te}\right)^m M_X(t),</math>
For any <math>X,m\ge 0</math> and <math>t>0</math>.

This follows from the inequality <math>1+x\le e^x</math> into which we can substitute <math>x'=tx/m-1</math> implies <math>tx/m\le e^{tx/m-1}</math> for any {{nowrap|<math>x, t, m \in \mathbb R</math>.}}
Now, if <math>t > 0</math> and <math>x,m\ge 0</math>, this can be rearranged to <math>x^m \le (m/(te))^m e^{tx}</math>.
Taking the expectation on both sides gives the bound on <math>\operatorname{E}[X^m]</math> in terms of <math>\operatorname{E}[e^{tX}]</math>.

As an example, consider <math>X\sim\text{Chi-Squared}</math> with <math>k</math> degrees of freedom. Then from the [[Moment-generating function#Examples|examples]] <math>M_X(t) = (1-2t)^{-k/2}</math>.
Picking <math>t=m/(2m+k)</math> and substituting into the bound:
<math display="block">\operatorname{E}[X^m] \le {\left(1 + 2m/k\right)}^{k/2} e^{-m} {\left(k + 2m\right)}^m.</math>
We know that [[Chi-square distribution#Noncentral moments|in this case]] the correct bound is <math>\operatorname{E}[X^m]\le 2^m \Gamma(m+k/2)/\Gamma(k/2)</math>.
To compare the bounds, we can consider the asymptotics for large <math>k</math>.
Here the moment-generating function bound is <math>k^m(1+m^2/k + O(1/k^2))</math>,
where the real bound is <math>k^m(1+(m^2-m)/k + O(1/k^2))</math>.
The moment-generating function bound is thus very strong in this case.