First af all, you should calculate:

a) P(X_n is a record)=?

then:

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_i’s 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.