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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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In case where $N$ has a Poisson distribution and is independent of $X_i$'s, it is called a compound Poisson random variable. – Sangchul Lee Sep 20 '12 at 9:36
It is interesting and reminds me sequential detection. Does it have only mathematical importance or used for some sort of modeling of some physical phenomena? – Seyhmus Güngören Sep 20 '12 at 10:19
@Seyhmus: it models something fairly natural for actuaries. Imagine 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. – Qiaochu Yuan Sep 20 '12 at 17:18
up vote 6 down vote accepted

It is called compound random variable. Please check:

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