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Difficulty: Medium
Category: Probability & Statistics
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Topics: probability, geometric-distribution, negative-binomial-distribution, sums-of-random-variables
You are modeling the number of trades it takes for a certain event to occur on $r$ independent assets. Assume that the number of trades, $X_i$, until the event occurs for asset $i$ follows a Geometric distribution with success probability $p$, i.e., $X_i \sim Geometric(p)$ for $i = 1, 2, ..., r$. The $X_i$ are independent. What is the distribution of the total number of trades, $S = X_1 + X_2 + ... + X_r$, until the event occurs on all $r$ assets?
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