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here's a question I got for homework (sorry if my translation is a bit unclear):

Let $X\sim‬G(p_1)$, $Y\sim ‬G(p_2)$, $X$ and $Y$ are independent. Prove that the minimum is also geometric, meaning: $\min(X,Y)\sim G(1-(1-p_1)(1-p_2))$.

Instructions: first calculate the probability $P(\min(X,Y) > k)$ and compare it to the parallel probability in (of?) a geometric random variable.

I have no idea where to start, even with the great clue that they've supplied. Any hints?

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    I found several later posts of exactly the same question. Judging by the content, it's hard to decide which one should be deemed the original and the others duplicate. In chronological order: [845706](https://math.stackexchange.com/questions/845706), [1040620](https://math.stackexchange.com/questions/1040620), [1056296](https://math.stackexchange.com/questions/1056296), [1169142](https://math.stackexchange.com/questions/1169142), and [1207241](https://math.stackexchange.com/questions/1207241).2018-03-21

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Let $X$ and $Y$ be independent random variables having geometric distributions with probability parameters $p_1$ and $p_2$ respectively. Then if $Z$ is the random variable $\min(X,Y)$ then $Z$ has a geometric distribution with probability parameter $1-(1-p1)(1-p2)$.

There are essentially two ways to see this:

First, the method outlined by the hint in your homework - Note that the cdf of $X$ is $1-(1-p_1)^k$ and the cdf of $Y$ is $1-(1-p_2)^k$, so the probability that $X>k$ is $(1-p_1)^k$ and the probability that $Y>k$ is $(1-p_2)^k$ and so the probability that both are greater than $k$ is $\left[(1-p1)(1-p2)\right]^k$. But the probability that both are greater than $k$ is the same as the probability that the minimum of the two is greater than $k$. From this we can get the cdf of $Z$ as $1-\left[(1-p1)(1-p2)\right]^k$, and we can note that this is the cdf of a geometric random variable with probability parameter $1-(1-p_1)(1-p_2)$.

Second, and more intuitively to me, we can go back to the definition of a geometric random variable with probability parameter $p$ : the number of Bernoulli trials with probability $p$ needed to get one success. So $\min(X,Y)$ is the number of trials of simultaneously running a Bernoulli experiment with probability $p_1$ and one with probability $p_2$ before one or the other experiments succeeds. The probability of one of the two experiments succeeding at any step is just $1-(1-p_1)(1-p_2)$, so $Z$ is a geometric random variable with probability parameter $1-(1-p_1)(1-p_2)$.

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    A minor correction to my comment: in the first line, for "the probability of", please read "the event"2011-12-13