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Let $X$ be a random variable and let $N$ be a discrete random variable which takes values in the non-negative integers. Let $X_1, X_2, ...$ be a sequence of i.i.d. random variables with the same distribution as $X$, all of which are also independent of $N$. Is there a name for the random variable

$Y = X_1 + X_2 + ... + X_N?$

The only hint I've found is that this appears to be what actuaries call the aggregate risk model. One reason I ask is that there is a very nice expression for the cumulant generating function $C_Y$ of $Y$ in terms of the cumulant generating functions $C_X, C_N$ of $X, N$, namely

$C_Y = C_N \circ C_X.$

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    @Seyhmus: it models something $f$airly natural for actuaries. I$m$agine you have some known probability distribution for bad things happening per unit time (e.g. car crashes) and also some known probability distribution for how much each bad thing costs you; then this construction describes the distribution of your total cost due to bad things happening.2012-09-20

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It is called compound random variable. Please check:

http://en.wikipedia.org/wiki/Panjer_recursion