Geometric Distribution problem

Question

Home work assignment i'm struggling with , i've been on this for an hour and have nothing :

Let $X_1, \ldots X_n$ be a group of independent variables with geometric distribution: $X_i \sim \operatorname{Geom}(p_i)$. Let $Y = \min \{X_1 , \ldots , X_n\}$.

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 $n\geqslant0$.