# Expectation of discrete random variable

Give a sequence of random variables $x_1,..,x_n$ with $x_n$ having a density of:

$$f_N(x) = \begin{cases} \frac{2N-1}{3N};x=1\\ 1/3;x=1+\frac{1}{N+1} \\ \frac{1}{3N};x=2\end{cases}$$

What would be the expectation and to which value does this function converge to?

I'd assume that this function converges in mean square to a fixed value and is degenerate, but i'm not quite sure what the expectation would be:

$$\lim_{N \to \infty} E(X) = \frac{2N-1}{3N} + 1/3*(1+\frac{1}{N+1})+2*(\frac{1}{3N})$$

As far as i can see this converges to 2/3 but that seems wrong.

• It looks like 2/3 + 1/3, no? A little nit: your $f$ is a probability mass function, not a density. – Adrian Mar 26 '16 at 14:08