# Tagged Questions

Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-...

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### statistics- probability question [closed]

Let E be the event that a corn crop has an infestation of ear worms, and let B be the event that a corn crop has an infestation of corn borers. Suppose that P(E) = 0.24, P(B) = 0.16, and P(E and B) =...
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### $(B_t)_{t\ge 0}$ be Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t}$ is an analytic function. [on hold]

Let $(B_t)_{t\ge 0}$ be a one-dimensional Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t} \; \text{for all} \; t\ge 0, \xi \in \mathbb{R}$ is an analytic function. A more general question ...
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### Irwin - Hall distribution of n different uniform distributions $U_k(a_k,b_k)$

Given $n$ independent and identically distributed uniform distributions $U(0,1)$, their sum is: $$X=\sum_{k=1}^nU_k(0,1)$$ The $pdf$ of this sum is given by the well known Irwin - Hall distribution. ...
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### Why Characteristic function is given such a name?

I've come to know that, For random variable $X$ ,with a Probability mass function P, $\phi_X(t)$ defined by : $\phi_X(t) : \mathbb R \to \mathbb C$ $\phi_X(t) = E(e^{itX}) =E[\cos tX + i\sin tX]$ ...
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### How to prove that if $E[X^2]$ is finite then $n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0$?

Let $X$ be a random variable with $E[X^2]<\infty$. I want to prove that $$n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0 \text.$$ I tried to apply Chebyshev's inequality, ...
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### Given $n$ heads out of $n$ tosses. What is the posterior probability that coin is fair? [closed]

I am given an $\sigma$-fair coin with the probability of head $(\theta)$ being in the interval $[\frac{1}{2} - \sigma, \frac{1}{2} + \sigma]$. Also I am given: For a Bayesian analysis of the ...
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### Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
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### $\{f_n\}$ is uniformly integrable if and only if $\sup_n \int |f_n|\,d\mu < \infty$ and $\{f_n\}$ is uniformly absolutely continuous?

Let $(X, \mathcal{A}, \mu)$ be a measure space. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon$ there exists $M$ such that\int_{\{x : |f_n(x)| > M\}} |f_n(x)...
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### How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ ...
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### Bernoulli Trials: Law of Large Numbers vs Gambler's Fallacy, the N paradox

I have asked this question before but I think it wasn't clear what I implied with my succinct question, so I will be a bit more verbose this time. Lets set the following example: Bernoulli trials, K=...
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### Use Bayes rule to test whether patient has disease after several positive tests

I have solved one of those standard bayes rule application exercises a la: Given a prevalence value of a disease, the sensitivity and the specificity of a test, calculate the probability that the ...
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### Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...
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### Infinite dimensional Borel-measurable function.

I am not quite sure how this statement about infinite dimensional borel-measurable functions is true, but the author says it is an easy observation: Let $D([0,\infty))$ denote the space of all ...
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### How to physically model/construct a biased coin?

A perfectly unbiased coin is one that has the same probability for heads and tails (i.e., 50%/50%). A perfectly biased coin is one that has (as the name suggests) different probabilities for head ...
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### When is a stochastic integral a martingale?

In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded ...
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### Why is a continuous Lévy process twice integrable?

In his textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008), Jochen Wengenroth shows (p. 144) that if $(X_t)_{t\in[0,\infty)}$ is a continuous, real-valued Lévy process with $X_t\in \mathcal{L}_2$ ...
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### How to compute integral of exponential martingale with respect to Poisson process?

Let $N=\{N_t:t\in\mathbb R_+\}$ be a homogeneous Poisson process with intensity $\alpha$ and $M_t=N_t-\alpha t$ the compensated process. I'd like to show that $N$ is not a natural process, i.e. that ...
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### Asymptotics of $\lim\inf X_n$, missing part of exam question [closed]

So I was working through an old exam and encountered the hilarious situation that part of the statement of the exercise was illegible. I was wondering if anyone could figure it out for me, so that I ...
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### Are martingales progressively measurable? (Application to square integrable martingales)

This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively ...
I have encountered an exercise that was quite puzzling for me. Maybe someone can help me out here? So let $(X_n)_n$ be $N(-a,1)$ distributed, independent random variables where $a>0$. I need to ...
Proof Attempt: For any random variable $X$ with non-decreasing continuous cdf $F(x)=\Pr(X≤x)$ (note that the inverse function does not necessarily exist due to flat regions in $F$), I wish to prove ...