# Tagged Questions

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### Central value of the partial exponential function [duplicate]

I need help calculating the central value of the partial exponential function : $$\lim_{n \to \infty} e^{-n} \sum^n_{k=0} \frac{n^k}{k!}$$ fd
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### Independence is preserved under taking almost sure limits

I´m not sure why this theorem is right, how can i prove it? Let $X_1,X_2....$ and $Y_1,Y_2...$ be random variables such that $X_n,Y_n$ are independent for every $n∈\mathbb N$ and such that X, resp ...
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### What is the limit of a sequence of events? Probability

Here's a question I'm sturglling with: Show that for an increasing sequence of events $$A_1\subset A_2\subset A_3\subset ...$$ the next equation holds ...
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### Limit of sum of indicator function

I came across a problem involving the following limit: $\lim_{n \to \infty} (\frac{1}{n} \sum\limits_{i=1}^n \mathbf{1}_{x_i>0}), \mbox{ where } X \sim N(\mu, \sigma)$ How would you approach ...
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### Convergence to the stable law

I am reading the book Kolmogorov A.N., Gnedenko B.V. Limit distributions for sums of independent random variables. From the general theory there it is known that if $X_i$ are symmetric i.i.d r.v ...
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### Exercise: Limits and Probability Measure

Let $\mu$ be a probability measure on $X$ (closed but unbounded), so that $\int_X \mu(dx) = 1$. Let functions $f_i:X \rightarrow \mathbb{R}_{\geq 0}$, $i = 1,2,...$, be Uniformly Integrable. Prove ...
Let $\mu$ be a probability measure over the (closed but unbounded) set $X \subseteq \mathbb{R}^m$: $\int_X \mu(dx) = 1$. Consider function $f:\mathbb{R}^n \times \mathbb{R}^n \times X \rightarrow ... 2answers 1k views ### Transform uniform distribution to normal distribution using Lindeberg–Lévy CLT Currently i am developing a game which is based on many computations of random values and therefore i have implemented many algorithms like the Mersenne-Twister etc. Unfortunately, all generators ... 1answer 378 views ### Fatou's Lemma and Almost Sure Convergence (Pt. 1) I have a question regarding Fatou's Lemma and a sequence of random variables converging almost surely. Fatou's Lemma states If$\forall n \in \mathbb{N}, \,\, X_{n} \ge 0$and$\displaystyle X = ...
I've been fighting with this homework problem for a while now, and I can't quite see the light. The problem is as follows, Assume random variable $X \ge 0$, but do NOT assume that ...