Using the fact that you can show for all by induction.

Using the same fact, for the variance you can derive the following recurrence:

(hopefully I have not made any mistakes in my calculations)

The solution to the recurrence (using generating functions for example) is

.

Note I didn't actually solve the recurrence myself, I used wolfram alpha.

so there may be an error there.

Now that you know the true expectation and variance, maybe try altering your guess to a normal with those values? Tell me if it seems more reasonable. To prove the actual distribution is normal with that expectation and variance shouldn't be too hard if its true ( I just might bite my tongue on that later .)

Whats interesting to note is that

.