A pair of fair dice is tossed. Let the random variable X is defined as X(a,b)=max⁡(a,b) Find the probability function of the r.v. X.

It’s clear that:
S={(1,1),…………,(6,6)}  , and
X={1,2,3,4,5,6}
∴P(X=1)=P{1,1}=1/36
   P(X=2)=P{(1,2),(2,1),(2,2)}=3/36
  P(X=3)=P{(1,3),(3,1),(2,3),(3,2),(3,3)}=5/36
  P(X=4)=P{(1,4),(4,1),(2,4),(4,2),(3,4),(4,3),(4,4)}=7/36
  P(X=5)=P{(1,5),(5,1),(2,5),(5,2),(3,5),(5,3),(4,5),(5,4),(5,5)}=9/36
  P(X=6)        =P{(1,6),(6,1),(2,6),(6,2),(3,6),(6,3),(4,6),(6,4),(5,6),(6,5)          =11/36

It’s clear that:   0≤P(x)≤1  ∀x  
And     ∑_x〖P(x)=1/36 [1+3+5+7+9+11]=36/36=1〗