Say a point is selected at random inside a square: {(x,y): 0<=x<=beta,0<=y<=beta}. It can be assumed that any point in the square is equally likely. Let Z be the random variable who is given by the min of the two coordinates of the randomly selected point in the square. What is the sample space of S and of Z? Also what is the cdf of Z.

 

You forgot to define S.

Also, please expand the abbreviation cdf. Mathematics is a universal language, but the words describing them are not.