Convergence in distibution of a product

I'm having afew troubles with the following questions, any help would be greatly appreciated.

**Q1.** Suppose X1,X2,...,Xn are iid random variables, each uniformly distributed on the interval [0,1]. Calculate the mean and variance of log(X1).

- i think i've done this although i can't help but get a variance of -3 ? getting a mean of -1.

suppose that 0=< a =< b, show that

**P((X1X2....Xn)^1/sqrt(n) in [a,b])** tends to a limit as n tends to infinity and find an expression for it.

-I'm not sure what to do for this, maybe something to do with using the log, as the log of a product is the sum of the logs..

**Q2. Poisson sampling**

Suppose X1,X2,...,Xn are iid random variables, each with the poisson distribution of parameter lambda>0.

Let X' =(X1+X2+...Xn)/n, find the support and pmf of X'.

- Now i obviously know the pmf of X1,X2 etc.. but not quite sure what to do when you sum random variables. my idea was to add the pmf of all the Xi's, but use k=k/n instead. where pmf=P(X=k).

any help? thanks alot