Editing Moment-generating function (section)

==Important properties==

Moment generating functions are positive and [[Logarithmically convex function|log-convex]],{{Citation needed|reason=log-convexity|date=June 2023}} with ''M''(0) = 1.

An important property of the moment-generating function is that it uniquely determines the distribution. In other words, if <math>X</math> and <math>Y</math> are two random variables and for all values of&nbsp;{{mvar|t}},

<math display="block">M_X(t) = M_Y(t), </math>
then
<math display="block">F_X(x) = F_Y(x) </math>

for all values of {{mvar|x}} (or equivalently {{mvar|X}} and {{mvar|Y}} have the same distribution). This statement is not equivalent to the statement "if two distributions have the same moments, then they are identical at all points." This is because in some cases, the moments exist and yet the moment-generating function does not, because the limit

<math display="block">\lim_{n \to \infty} \sum_{i=0}^n \frac{t^i m_i}{i!}</math>

may not exist. The [[log-normal distribution]] is an example of when this occurs.
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If the moment generating function is defined on such an interval, then it uniquely determines a probability distribution. -->

===Calculations of moments===
The moment-generating function is so called because if it exists on an open interval around {{math|1=''t'' = 0}}, then it is the [[exponential generating function]] of the [[moment (mathematics)|moments]] of the [[probability distribution]]:

<math display="block">m_n = \operatorname{E}\left[ X^n \right] = M_X^{(n)}(0) = \left. \frac{d^n M_X}{dt^n}\right|_{t=0}.</math>

That is, with {{mvar|n}} being a nonnegative integer, the {{mvar|n}}-th moment about 0 is the {{mvar|n}}-th derivative of the moment generating function, evaluated at {{math|1=''t'' = 0}}.