Moment-generating functions (MGFs) are mathematical functions that provide a way to encapsulate all the moments of a probability distribution. They are defined as the expected value of the exponential function of a random variable, specifically expressed as $M_X(t) = E[e^{tX}]$, where $X$ is a random variable and $t$ is a real number. MGFs are particularly useful in probability theory because they can be used to derive properties of distributions, such as means and variances, and facilitate the computation of sums of independent random variables.
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