Minimum of two geometric random variables is geometric [duplicate]
Solution 1:
Use distribution function:
$$P(\min(X,Y) \le x) = 1 - P(\min(X,Y) \ge x) = 1 - P(X \ge x, Y\ge x) = 1 - P(X \ge x) P(Y \ge x)$$
Now compute that and recognise the distribution function of a geometric distribution