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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 \... • 6,785 0 votes Accepted ### 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 ... 0 votes ### 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 ... • 158k 0 votes ### 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 ... • 158k -1 votes ### 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 ... 1 vote Accepted ### 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) \\&... • 2,739 1 vote Accepted ### 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) \...
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Just replace $h\theta$ by $t$ and use $\sum_{n=0}^{q-1}r^n=\frac{r^q-1}{r-1}$ for $r=e^{2i\pi t}.$