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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, ...
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
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
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
$$I_3=\int_{-1}^1\int_{-1}^1 \int_{-1}^1 |x+y+z|\sqrt{1-x^2}\sqrt{1-y^2}\sqrt{1-z^2}\,dx\,dy …
- 3,255
3
votes
Accepted
How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\si...
Call the left resp. right hand sum $R_m$ resp. $R_{m+1}$. As $L$ is $\exp(\lambda)$ and independent of $(R_m,R_{m+1})$ , taking expection with resp. to $L$ first gives
$$\mathbb{P}(R_m\leq L < R_{m+1 …
- 3,255