Probability mass function of $M=\min\{X_¡: 1≤i≤ n\}$.

Question

Let $X_1, X_2, \cdots, X_n$ be independent random variables with $X_i$ having probability mass function

$$ P(X_i = k) = \left(\frac{i}{i+1}\right)^k \frac{1}{i+1} $$

for all $k = 0, 1, 2, \cdots$ and for all $i = 1, 2, \cdots, n$. Let

$$M=\min\{X_i: 1 \leq i \leq n\}.$$

Then derive the probability mass function of $M$. Please suggest me how to find me the $\text{pmf}$ of $M$.

Answer 1

Note that if $X\sim\mathrm{Geo}(p)$ and $Y\sim\mathrm{Geo}(q)$ are independent, then for each nonnegative integer $k$, \begin{align} \mathbb P(X\wedge Y>k) &= \mathbb P(X>k,Y>k)\\ &= \mathbb P(X>k)\mathbb P(Y>k)\\ &= (1-p)^{k+1}(1-q)^{k+1}, \end{align} so $$ \mathbb P(X\wedge Y\leqslant k) = 1 - ((1-p)(1-q))^{k+1}, $$ and hence $X\wedge Y\sim\mathrm{Geo}(1-(1-p)(1-q))$. It follows that $$\bigwedge_{i=1}^n X_i \sim\mathrm{Geo}(1-\rho),$$ where \begin{align} \rho= \prod_{i=1}^n \frac i{i+1} = \frac1{n+1}, \end{align} that is, $$ \mathbb P\left(\bigwedge_{i=1}^n X_i = k\right) = \left(\frac1{n+1}\right)^k\frac n{n+1}. $$