A moment-generating function (MGF) is a mathematical tool used in probability theory to summarize all the moments of a random variable. It is defined as the expected value of the exponential function of the random variable, typically expressed as $M_X(t) = E[e^{tX}]$. This function not only helps in finding moments like mean and variance but also plays a key role in connecting continuous probability distributions and partition functions through generating functions.
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