Find the moments of the random variable $X$ if its moment generating function is $$M_X(t) = (1−p_1 −p_2)+p_1e^t +p_2e^{2t}.$$ What is the variance of $X$?
2026-03-29 22:15:17.1774822517
What is the moment generating function and variance
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Hint: The moments of $X$ can be obtained by differentiation of $M_X$: $$ {\rm E}[X^n]=M^{(n)}_X(0),\quad n\in\mathbb{N}. $$ The variance of $X$ can be calculated once you've found ${\rm E}[X]$ and ${\rm E}[X^2]$ using a well-known formula.