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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?

user352102
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  • You should read the Law of Large Numbers and Central Limit Theorem carefully. – angryavian Feb 26 '18 at 23:28
  • This is not a Riemann sum. Both questions seem to be applications of the law of large numbers, the second with random variable $Y_i = 1_{c(X_i)\neq 1}$. Of course your notation $p(c(x)\neq 1)$ as a PDF is not good notation, you should just use expectations $E[X]$ and $E[Y]$, and computing expectations of a 2-valued random variable is best done by using the probability mass function. – Michael Feb 26 '18 at 23:54
  • I also observe that "sampled from a uniform distribution on the real line $\mathbb{R}$" does not make sense, unless you mean a particular (finite) interval of the real line. There is no such thing as a random variable uniformly distributed over all the reals. It is also not clear how you arrive at the sentence with "from the normal distribution": Note that a normal distribution usually means a Gaussian distribution, but no Gaussian assumptions were given on $X_i$, rather, it was claimed those were uniformly distributed. – Michael Feb 26 '18 at 23:58

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