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I'm studying statistics using Rice's Mathematical Statistics and Data Analysis.

We can find the $r$th moment of a random variable $X$ by taking the moment generating function $M(t)$ of $X$ and taking the derivative $r$ times. Then substituting $t=0$, we arrive at the $r$the moment. Are moment generating functions significant in any way for values $t \neq 0$?

Snowball
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    Yes. For one, the derivative at zero depends on the values of the MGF in a neighborhood of zero, not just its value at zero. – passerby51 Feb 02 '21 at 00:39
  • See https://stats.stackexchange.com/questions/304066/real-life-uses-of-moment-generating-functions/304069#304069 for examples! – kjetil b halvorsen Feb 02 '21 at 01:33

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