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Results tagged with integration
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user 48831
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
4
votes
Expected absolute value of the average of two points from the disc
(Too long for a comment)
For the record:
$$\mathrm{exp\_abs}(3)=\frac{4}{3 \pi^2}\,I_3=0.3671989447$$
where
$$I_3=\int_{-1}^1\int_{-1}^1 \int_{-1}^1 |x+y+z|\sqrt{1-x^2}\sqrt{1-y^2}\sqrt{1-z^2}\,dx\,dy …
- 3,255
14
votes
Accepted
Integral $\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}...
Here is another approach, which also gives the rational term.
(I) To see how it works let $n\geq 2$ and consider first
the simpler case
\begin{align*}
\mathbb{E}\bigg(\frac{1}{X_1+\ldots+X_n}\bigg)= …
- 3,255
24
votes
Binomial again, and again
Here is a proof which doesn't use the identity $\int_{-\infty}^\infty {n \choose x}\,dx= 2^n$:
Using the representation ${ n \choose x}=\frac{1}{2\pi}\int_{-\pi}^\pi e^{-ixt}\left(1+e^{it}\right)^n\, …
- 3,255
3
votes
Accepted
How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\si...
Call the left resp. right hand sum $R_m$ resp. $R_{m+1}$. As $L$ is $\exp(\lambda)$ and independent of $(R_m,R_{m+1})$ , taking expection with resp. to $L$ first gives
$$\mathbb{P}(R_m\leq L < R_{m+1 …
- 3,255