May $Z$ be a random variable distributed as $N(0,1)$ Find the following limit: \begin{equation} \lim_{n\to\infty}\mathbb{E}\left[\frac{1}{\sqrt{n}-Z}\right] \end{equation}
How does one go about proving it?
May $Z$ be a random variable distributed as $N(0,1)$ Find the following limit: \begin{equation} \lim_{n\to\infty}\mathbb{E}\left[\frac{1}{\sqrt{n}-Z}\right] \end{equation}
How does one go about proving it?
The expected value does not exist: the function $ \dfrac{f(z)}{\sqrt{n}-z}$ (where $f$ is the probability density function) is not absolutely integrable because of the singularity at $z=\sqrt{n}$. However, the Cauchy principal value of this improper integral does exist. That is, if $I_r(t) = 1$ for $|t|\ge r$ and $0$ for $|t|