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**noob mathematician** Let $\displaystyle P(X=i)=P(Y=i)=\frac{1}{2^i}$

Where $\displaystyle (i=1,2,...)$.

X and Y are independent.

Then how can I find $\displaystyle P(min(X,Y)\le i)$.

I know that when $\displaystyle P(max(X,Y)\le i)$ means that both X and Y are less than i so I can conclude that $\displaystyle P(X\le i,Y\le i)=P(X\le i)P(Y\le i)$. But how can I do the same for Min?

Meanwhile, since this is a discrete distribution, I am also facing the problem of finding its mass probability function. e.g. $\displaystyle P(X\le i)=\sum_{X_n<i} p(X_n)$ to solve for some other parts. Any suggestion?

TIA