Slutsky and expected value

Hi,

I am still hoping to follow up on this. Maybe I'll post what I did and you could point out where I went wrong?

Take iid sample of random variables that are Exp(1)

b) Find $\displaystyle E(\bar Y_n)^2 $

I take that $\displaystyle \bar Y_n \to \mu = 1 $ in probability.

I then can use Slutsky with $\displaystyle g(y) = y^2 $

to give that $\displaystyle g(\bar Y_n) \to g(\mu) $ in distribution

and thus $\displaystyle (\bar Y_n)^2 \to \mu^2 = 1 $ in distribution and probability.

Then $\displaystyle E(\bar Y_n)^2 = \mu^2 = 1 $

What am I missing here???

Thanks a ton!