First af all, you should calculate:
a) P(X_n is a record)=?
b) E [number of records by time n]=?
c) Var [number of records by time n]=?
I give you some hints how to solve these problems:
For part (a) use symmetry and observe that P(X_i = X_j ) = 0
since X_iís are continuous random variables. For parts (b) and (c) introduce indicator random variables. Note that these indicator random variables are independent
in this case.
I hope I helped you.