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Two faulty machines, M1 and M2, are repeatedly run synchronously in parallel (i.e., both machines execute

one run, then both execute a second run, and so on). On each run, M1 fails with probability p1 and M2 with

probability p2, all failure events being independent. Let the random variables X1, X2 denote the number of

runs until the ﬁrst failure of M1, M2 respectively; thus X1, X2 have geometric distributions with parameters

p1, p2 respectively.

Let X denote the number of runs until the ﬁrst failure of either machine. Show that X also has a geometric

distribution, with parameter p1 + p2 − p1p2

Attempt at an answer:

X1 has a geometric distribution of (1-p1)^i-1 * p1

X2 has a geometric distribution of (1-p2)^i-1 * p2

I'm confused an don't know how to proceed. Any help is appreciated.

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# Geometric Distribution

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