Intereting Posts

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Are there broad or powerful theorems of rings that do not involve the familiar numerical operations (+) and (*) in some fundamental way?
Solving a Recurrence Relation/Equation, is there more than 1 way to solve this?
Conceptualizing Inclusion Map from Figure Eight to Torus
Weird $3^n$ in an identity to be combinatorially proved
Question Relating with Open Mapping Theorem for Analytic Functions
Formula of a sum
The Three Princesses (distinguishing truth-teller with 1 question)
The Line with two origins

May you have an idea for the following exercise I found from some olympiad.

Each day you have to bring home one full can of water. To do so you go to the next well and make the can full. On the way home you loose a proportion which is uniformly distributed on [0,1].

Question: How many times, on average, do you need to go and get water each day?

- If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $
- Proof $ \int_0^\infty \frac{\cos(2\pi x^2)}{\cosh^2(\pi x)}dx=\frac 14$?
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Intuitively I would guess the answer is 2 but I do not know how to show this formally.

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Suppose $U_k$ is the amount of water you bring during $k$-th visit to the well, and $\{U_k\}$ are iid uniform random variables on $(0,1)$. Let $N$ denote the random variable of the number of trips to the well needed to bring home a full can.

Then

$$

\{N=n\}= \{ U_1+\cdots+U_{n-1} < 1, U_1+\cdots+U_{n-1}

+ U_n \geqslant 1 \}

$$

That is

$$

\mathbb{P}\left(N=n\right) = \mathbb{P}\left( U_1+\cdots+U_{n-1} < 1, U_1+\cdots+U_{n-1}

+ U_n \geqslant 1 \right) = \frac{1}{n \cdot (n-2)!} \tag{$\ast$}

$$

The expected number of trips is thus:

$$

\mathbb{E}\left(N\right) = \sum_{n=2}^\infty n \frac{1}{n \cdot (n-2)!} = \mathrm{e} \approx 2.71828 > 2

$$

To prove $(\ast)$ note that $V_{n-1} = U_1 + \cdots + U_{n-1}$ follows Irwin-Hall distribution, and that

$$

\mathbb{P}\left(V_{n-1} < v \leqslant 1\right) = \frac{v^{n-1}}{(n-1)!}

$$

as a volume of the corner section of $n-1$-dimensional unit hypercube by the plane $u_1 + \cdots + u_{n-1} = v$. Therefore, $f_V(v) = \frac{v^{n-2}}{(n-2)!}$ for $0<v<1$.

$$

\mathbb{P}(N=n) = \mathbb{P}\left( V_{n-1} < 1, U_{n} + V_{n-1} > 1\right) =

\int_0^1 \int_{1-v}^1 \frac{v^{n-2}}{(n-2)!} \mathrm{d} v \mathrm{d}u = \frac{1}{n (n-2)!}

$$

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