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The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true. For example, it gives some evidence that there are finitely many ...

I am doing some exercises in the analytic and there is a problem as following: ``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that: $\sum\limits_{n=1}^{+\infty} f_n = 1$. $\...

Let $X$ be a standard normal random variable. What $\alpha$ minimizes $E|X|^{\alpha}$? Numerically, $\alpha$ turns out to be equal to $\sqrt{3}/2-\varepsilon$ where $\varepsilon$ is of the order $10^...