# 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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### Origin of the notation for statistical divergence

The unusual notation $D(P||Q)$ seems to be universally used for statistical divergences (e.g. KL divergence). What is the origin of this notation, and do the double bars (pipe symbols) have any ...
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### Autocorrelation function of a Wiener process & Poisson process.

Can anyone possibly explain step 3 and 4 in this solution?
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### Markov chain: Find expected value to get back to starting state

I wonder why they complicate this solution? Call the mean time to get from i to j $M_{i,j}$ and set up three simple equations starting with $$M_{0,0} = 1 + (1/3)M_{1,0} + (1/3)M_{2,0}$$ and you get ...
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### Expressing equal probability on an infinite line with probability axioms

Is there any way using the usual (Kolmogorov) axioms of probability to describe/model the following situation : A value $v \in \mathbb{R}$ has an equal probability of being measured anywhere in the ...
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### Constant independent random variables

How can I prove this : Let $X$, $Y$ be independent random variables and suppose $P(X +Y = α) = 1$, where $α$ is a constant. Show that both $X$ and $Y$ are constant random variables.
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### Does local martingale have the same mean value as well? [duplicate]

We know that if $\{M_n\}$ is a martingale, we know from definition of martingale that $E(M_n) = M_0$ for all $n \geq 0$. However, if we only know that a sequence of random variables $\{X_n\}$ is a ...
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### Why does probability need the Axiom of pairwise disjoint events? [duplicate]

I'm a beginning student of Probability and Statistics and I've been reading the book Elementary Probability for Applications by Rick Durret. In this book, he outlines the 4 Axioms of Probability. ...
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### Birth and death process. Total time spent in state $i$.

Question: Let $X(t)$ be a birth-death process with $\lambda_n = \lambda > 0$ and $\mu_n = \mu > 0,$ where $\lambda > \mu$ and $X(0) = 0$. Show that the total time $T_i$ spent in state $i$ is ...
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### Is $H$-measure actually monotonic (at least on hyperrectangles)?

I am currently reading Introduction to Copulas by R. B. Nelson. First chapter introduces some theory of 2-monotone functions and I am trying to extend it for $n$-dimensional hyperrectangles as an ...
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### Upper bound for random walk to show stopping time is bounded

I have a simple symmetric random walk (SSRW), and a stopping time: $\tau=\inf\{ n \geq 0 ~:~ |S_n|=N\}$. I am showing that $\newcommand{\ee}[1]{\mathbb{E}[#1]}$ $\newcommand{\pp}[1]{\mathbb{P}[#1]}$ ...