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Convergence of Step Functions Generated by Uniformly Distributed Random Points

For any $k\ge 1$, as $0\le 1 - k\delta<1$ we have [1]: $$\lim_{n\to \infty} (-1)^k{n+1\choose k}(1 - k\delta)^n=\lim_{n\to \infty}\frac{(-1)^kn^k(1 - k\delta)^n}{k!}=0.$$ Hence, $$\lim_{n\to \...
Amir's user avatar
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Probability that a uniformly distributed point lies within a triangle in the unit disc

I figured out the answer & will share it incase anybody is searching for something similar: a) the probability of W lying in the triangle is just $\frac{E(Area_{triangle})}{Area_{circle}}$ b) a ...
edster101's user avatar
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Resampling techniques to obtain uniform random variables.

Your triangular distribution has density $f_X(x)=\dfrac{h-|x|}{h^2}\mathbb I_{[|x|\le h]}$. There are various approaches. The simplest is: If $h>1$, you could resample by accepting each $x_i$ with ...
Henry's user avatar
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Expected Value - Uniform distribution over infinite interval

Let's answer the question with a proper prior distribution for packet lengths $f_X(x)=\mathbb P(X=x)$. The likelihood of no errors is proportional to $(1-\alpha)^x$ and so (using Bayesian methods) the ...
Henry's user avatar
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Trains arrive at a station

Check if there is any fallacy in it. let X be the random variable representing the number of minutes past 9 that the passenger arrives at the station. a)he has to wait for less than 6 minutes if he ...
Prabhakar Kumar's user avatar
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Discrete Convexity of a Function

Note that you can write that $Y\equiv Y_x=W_1+\dots+W_x$ where $W_i$ are i.i.d. Bernoulli($p$) random variables. Hence \begin{align} &R(x+1)+R(x-1)-2R(x)=\mathbb{E}\max(Z-W_1-\dots-W_{x+1},0) \\&...
van der Wolf's user avatar
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Exponential Moment of Uniform Random Vector

Since each $z_i$ are iid random variables, $$ \mathbb{E}(\exp(\lVert Z \rVert^2)) = \mathbb{E}\left(\exp\left(\sum_{k = 1}^d z_k^2\right)\right) = \mathbb{E}\left(\prod_{k = 1}^d e^{z_k^2}\right) \...
Thành Nguyễn's user avatar
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Trouble proving inequality regarding the modulus of a complex number sum in uniform distribution theorem

Just replace $h\theta $ by $t$ and use $\sum_{n=0}^{q-1}r^n=\frac{r^q-1}{r-1}$ for $r=e^{2i\pi t}.$
Letac Gérard's user avatar

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