Say $X_i$ are iid random variables sampled from a uniform distribution on the real line $\mathbb{R}$. (1/n)$\Sigma_{i=1}^n$ $X_i$ "converges" to $\int_{\mathbb{R}}$ xp(x)*dx where p(x) is the probability density function for the normal distribution. Is this true?
Similarly consider a function c: $\mathbb{R} \rightarrow$ {1,-1} and assume we have n random variables $X_i$ as above where we can make n arbitrarily large.
Does: (1/n)*$\Sigma_{i=1}^n$ $1_{c(X_i) \neq 1}$ converge to $\int_{\mathbb{R}}$ 1*p(c(x) $\neq$ 1) *dx and why?
Can we view the LHS as a Riemann sum?