On the distribution of the minimum of geometrically distributed random variables

Question

I am struggling with the following homework assignment. I have been on this for an hour and have nothing:

Let $X_1, \dots X_n$ be independent random variables with geometric distribution, $X_i \sim \operatorname{Geom}(p_i)$. Let $Y := \min \{X_1 , \dots , X_n\}$. How is $Y$ distributed?

Answer 1

Hints:

• A random variable $X$ is geometric with parameter $p$ if and only if $\mathbb P(X\geqslant k)=$$_______$ for every $k\geqslant0$.
• The random variable $Y=\min\{X_1,\ldots,X_n\}$ is such that $[Y\geqslant k]=\bigcap\limits_{i=1}^n[X_i\geqslant k]$ and the random variables $(X_i)_i$ are independent hence $\mathbb P(Y\geqslant k)=\displaystyle\prod\limits_{i=1}^n$$______$ for every $n\geqslant0$.