Let the numbers p and q in the interval (0, 1) be given. Assume that X ∼ Geometric(p) and Y ∼ Geometric(q) are independent random variables.
Show that Z = min{X,Y} is geometrically distributed and find the corresponding parameter.
How do i approach this, because i can't really remember how i'm supposed to show/determine that Z = min{X,Y} is geometrically distributed. I also have no idea about how i should find the corresponding parameter and frankly i'm not sure i understand what the corresponding parameter actually is.
Well i think i know that $$ \begin{align*} P( \min\{X,Y\}) &= P( \min\{X,Y\} > k) \\ &= P((X > k), (Y > k)) \\ &= P(X > k)P(Y > k) \\ &= P(X > k)^2 \end{align*} $$ After that i'm assuming i have to use the probability mass function (PMF) or the cumulative distribution function (CDF) for a geometric distribution, but i don't know which to use or how im supposed to show that it is a geometric distribution like this.
If $X\sim Geo(p)$ and $Y\sim Geo(q)$, then $\min\{X,Y\}\sim Geo(1-(1-p)(1-q))$. Think like that: You are looking at the first time of success from either $X$ or $Y$. The probability of both events not happening is $(1-p)(1-q)$. Therefore the probability of at least one success is $1-(1-p)(1-q)$.