Haiden Meyer

2022-10-03

The minimum of two independent geometric random variables
Let $X\sim ‬G\left({p}_{1}\right)$, $Y\sim ‬G\left({p}_{2}\right)$, X and Y are independent. Prove that the minimum is also geometric, meaning: $min\left(X,Y\right)\sim G\left(1-\left(1-{p}_{1}\right)\left(1-{p}_{2}\right)\right)$.
Instructions: first calculate the probability $P\left(min\left(X,Y\right)>k\right)$ and compare it to the parallel probability in (of?) a geometric random variable.

antidootnw

Step 1
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 $1-\left(1-{p}_{1}\right)\left(1-{p}_{2}\right)$.
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 $1-\left(1-{p}_{1}{\right)}^{k}$ and the cdf of Y is $1-\left(1-{p}_{2}{\right)}^{k}$, so the probability that $X>k$ is $\left(1-{p}_{1}{\right)}^{k}$ and the probability that $Y>k$ is $\left(1-{p}_{2}{\right)}^{k}$ and so the probability that both are greater than k is ${\left[\left(1-{p}_{1}\right)\left(1-{p}_{2}\right)\right]}^{k}$. 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 $1-{\left[\left(1-{p}_{1}\right)\left(1-{p}_{2}\right)\right]}^{k}$, and we can note that this is the cdf of a geometric random variable with probability parameter $1-\left(1-{p}_{1}\right)\left(1-{p}_{2}\right)$.
Step 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 $1-\left(1-{p}_{1}\right)\left(1-{p}_{2}\right)$, so Z is a geometric random variable with probability parameter $1-\left(1-{p}_{1}\right)\left(1-{p}_{2}\right)$

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