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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3
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107 views

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 ...
1
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0answers
19 views

Markov-like inequality for bias of 0-1 random variables

Let $S \subseteq \{0,1\}^d$ for some integer $d$ and let $X_s$ be a random variable over $\{0,1\}$ for every $s \in S$. Define $\omega = \frac{1}{|S|}\sum_{s\in S}\text{bias}(X_s)$, where $\text{bias}(...
4
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1answer
95 views

Absolute Continuity of the sum of two Cantor random variables

If we have two independent random variables each having a Cantor distribution is there an easy way to see that the distribution of their sum is not absolutely continuous? I am pretty sure that if ...
1
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1answer
43 views

Law of total expectation well-defined?

Wikipedia states that this is a special case of the law of total expectation click me. Given a partition $A_1,...,A_n$ of the outcome space, we have for a random variable $X$ that $E(X)=\sum_{i=1}^{...
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0answers
43 views

Autocorrelation function of a Wiener process & Poisson process.

Can anyone possibly explain step 3 and 4 in this solution?
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0answers
34 views

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 ...
3
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2answers
41 views

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 ...
2
votes
1answer
80 views

Conditional pdf of product of two exponential random variables

I have two independent random variables say $X$, $Y$. Both of them follow exponential distribution with parameter $λ$ i.e $X\sim λe^{−λx}$ and $Y\sim λe^{−λy}$. I want to find the pdf of $Z=XY$ given $...
1
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1answer
36 views

Distribution of $X\cdot Y +a\cdot X$ for $X,Y$ standardnormal

I am searching for the exact or asymptotic CDF of the rv $X\cdot Y +a\cdot X$ with the $X,Y$ independent standard normal rv's. Found nothing till now. Any hints? Thanks.
1
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3answers
29 views

Comparing results after performing W/ and W/O replacement on an experiment

In this is famous example in the probability theory, there are 6-Red, 4-Green, and 5-Blue balls in a bag. By calculating the probability with and without replacement for these three colors, we ...
1
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1answer
61 views

M/M/1 or M/M/n?

In a queuing systems with a single queue that receives $n$ poisson arrival streams with arrival rates $\lambda_1, \lambda_2, ...,\lambda_n$, and exponential service rates of $\mu_1,..., \mu_n$, we can ...
2
votes
1answer
53 views

Problem calculating MGF (expectation) using just the definition.

Assume $ (\Omega,\mathcal{F})=([0,1],[0,1]\cap\mathcal{B}(\mathbb{R})$ . Let $(X_j)_{j\geq1}$ be a sequence of independent random variables s.t. $\mathbb{P}(X_j=k)=\frac{1}{3}, k=0,1,2, j=1,2,..$ ...
0
votes
1answer
46 views

Almost sure convergence of sum of countable number of sequences

I have a countable number of infinite sequences of random variables, $(X_n^i)_{i=1}^\infty$. I know that each sequence converges almost surely: for all $i \geq 1$, $$ X_n^i \overset{\text{a.s.}}{\...
0
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2answers
37 views

Understating binomial and uniform distributions.

Speaking of probability distribution, Can someone kindly tell how and when I use binomial distribution and uniform distribution in real life situations? I understand their mathematical formulas but I ...
3
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0answers
45 views

Necessary and Sufficient Conditions for a CDF

This is an attempt to prove Theorem 1.5.3. in Casella and Berger. Note that the only things that have been proven are really basic set-theory with $\mathbb{P}$ (a probability measure) theorems (e.g., ...
0
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1answer
38 views

Convolution domains probability theory

Problem 1.4 here: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/unit-ii/quiz-2/MIT6_041SCF13_quiz02....
3
votes
1answer
68 views

Probability man women in a survey

In a survey, we asked $7$ men and $5$ women. Is randomly selected without replacement persons one by one until a man. Let $X$ be a random variable of the number of prints required. Determine ...
2
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0answers
51 views

Finding limiting distribution of $\sqrt n \frac{(S_n-5n)}{\sum_{i=1}^n X_i^2}$

Let $X_1, X_2,...$ be IID rv with $E[X_1] = 5$ and $Var[X_1] = 9$ Find the limiting distribution of $\sqrt{n} \frac{(X_1 + X_2 + ... + X_n - 5n)}{(X_1^2 + X_2^2 + ... + X_n^2)}$ as $n \rightarrow \...
0
votes
1answer
113 views

What is a “continuous modification”? And can we always modify an almost surely continuous process, such that every path is continuous?

Let's motivate the question by a classical result: Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ which ...
1
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1answer
54 views

About convergence in probability

A few days ago, I was introduced the concept of convergence in probability, after the almost sure convergence. I understood the almost sure convergence (I think): We have a sequence of random ...
3
votes
1answer
82 views

Is this process a martingale?

Given $X_t=\int_0^t s W_s dW_s$ and the process $M_t=X_t^3-\int_0^t X_sY_s ds$. Find $Y_t$ such that $M_t$ is a martingale. I started thinking that $X_t$ can be seen as: $dX_t=tW_tdW_t$ then by ...
1
vote
1answer
55 views

Find expected value of a dice experiment

The following experiment is performed, Roll a dice. If you stick to the outcome, then the final score is the number on the dice. The experiment ends here. If the experiment is performed $n$ times, ...
2
votes
3answers
39 views

Convergence of independent $\mathcal U {(n,n^2)}$ random variables?

What does this sequence of random variable's distribution converge to? The random variables are given as follows $$Y_n=\frac{X_n-n}{n^2}, \quad n=1,2,3, \dots$$ and $X_n\sim\mathcal{U(n,n^2)}$- a ...
2
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0answers
60 views

Distribution of sums of inverses of random variables uniformly distributed on [0,1]

If I have $N$ random variables (denoted below as $X_i$) with uniform distribution on the $x$-axis $X_i = \rm{rand}[0,1]$ then the sum $$ S_N = \frac{1}{N}\sum_i^N\frac{1}{2X_i-1} $$ seems to be a ...
2
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0answers
25 views

On the Gaussian Poincare inequality

Let $X$ be a standard normal random variable. Then, for any differentiable $f:\mathbb{R}\to\mathbb{R}$ such that $\mathbb{E}f(X)^2<\infty,$ the Gaussian Poincare inequality states that $$\mathrm{...
2
votes
2answers
60 views

Why is the integral $\int_0^1t\,dW_t$ a normal random variable?

Consider the random variable $X=\int_0^1t\,dW_t$, where $W_t$ is a Wiener process. The expectation and variance of $X$ are $$E[X]=E\left[\int_0^1t\,dW_t\right]=0,$$ and $$ Var[X]=E\left[\left(\int_0^...
0
votes
0answers
50 views

how to calculate expected utility for probability decision problem?

consider a decision problem: classifying $x$ as belonging to one of two classes $C_1, C_2$. there are prior probabilities for each class, $p(C_1), p(C_2)$ and likelihood probabilities for data given ...
0
votes
1answer
37 views

Proving Borel Strong Law of Large Numbers using Bienaymé-Tchebichev inequality.

While reading Loeve's book on Probability (page 246, 5th edition) I found the following statement: It is of some interest to observe that Borel's law of large numbers can also be obtained by ...
2
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0answers
34 views

Why does the minimum exist in optimal transport?

Let $P,Q$ be any two distributions and let $\mathcal{M}(P,Q)$ be the set of all couplings of $P$ and $Q.$ For a given metric $d(\cdot,\cdot),$ the optimal transport cost is: $$\min_{(X,Y)\sim M\in \...
3
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0answers
41 views

Bayesian information criterion from measure theoretic point of view?

Bayesian information criterion (BIC) is well known and it is derived from the maximizing the posterior density function which is equivalent to solving the marginal likelihood integral. My question is: ...
2
votes
0answers
25 views

$p$th Variation

Let $f:[a, b]\rightarrow \mathbb{R}$. When we speak of total variation, every function must have total variation that exists and is nonnegative, although it may be $+\infty$. Let $p>1$. Define ...
1
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0answers
108 views

Kolmogorov's zero-one law

When reading this post (which is in my opinion one of the best posts on MSE by far), I read that the writing of Hamlet is a tail event, thus by Kolmogorov's zero-one law, it has a probability of ...
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0answers
93 views

Can you define a sensible probability measure on the set of countable transitive models of ZFC?

It is well known that the set of hereditarily countable sets $H(\omega_1)$ —or, if you prefer, $H_{\omega_1}$— has cardinality $2^{\aleph_0}$, and I understand that every countable ...
0
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0answers
48 views

Chernoff bound on expected absolute deviation

I have an exercise I can't quite solve and I'm starting to feel kind of stupid: Suppose we roll a fair dice with faces {1,2,3,4} 100 times. Let the random variable $X$ be the sum of the numbers that ...
2
votes
1answer
30 views

An example for Loeve's extension theorem on independence.

I have the following theorem from Loeve's book on probability: Extension theorem: Minimal $\sigma$-fields over independent classes $C_{t}$ closed under finite intersections are independent. I'...
2
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0answers
41 views

Quasistationary distribution for the Moran model

The Moran model is a model for genetic drift. Basically, it is a finite Markov chain (more precisely, a birth-death chain) with state space $S:=\{0,...,N\}$ and the following transition probabilites: ...
2
votes
1answer
27 views

Calculating a (Forward Measure) Martingale

Above is my question. I am, unfortunately, stuck on part (a)! Below are my workings. I've just spotted a typo -- at one point, an "$\exp$" is missing, but it's fairly obviously supposed to be there. ...
4
votes
0answers
39 views

What is a moment for a joint distribution?

I am currently self-learning probability theory using measure theory. Moments are defined as $\int X^k \ d P$, where $X$ is a stochastic variable and $P$ is the corresponding probability measure. If ...
1
vote
2answers
61 views

Autocorrelation function of integral of cont. white noise

Let $W(t)$ be continuous time white noise, that is, a wide-sense stationary (WSS) zero-mean Gaussian process with autocorrelation function $R_W (\tau) = σ^2\delta(\tau)$. Calculate the auto ...
0
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2answers
62 views

Discrete vs Continuous vs Random Variables

Can someone provide me a clear explanation (Perhaps with examples) about the difference between the following mathematical three concepts in probability: Discrete ...
1
vote
1answer
47 views

Weak Law of large numbers involving a sequence and random variable

During one of our Information theory classes, the professor constructed the following set: $$T_\delta = \left\{\mathbf{y} \in \mathbb{R}^n: \frac{\sum_{i=1}^ny_i^2}{n} \leq P + \sigma^2+\delta\right\}...
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0answers
36 views

Book on limit theorems for sums of random variables

I'm currently studying for an advanced course of probability theory and I would want to find a book which explains clearly limit theorems for sums of random variables. We used characteristic ...
1
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1answer
62 views

What is the “distributional derivative” of a Brownian motion?

Let $\emptyset\ne I\subseteq\mathbb R$ be an open interval and $A:C_0^\infty(I)\to\mathbb R$ be a distribution. Then, $$\langle{\rm D}A,\varphi\rangle:=-\langle A,\varphi'\rangle\;\;\;\text{for }\...
0
votes
1answer
170 views

What is the distribution of the limit of random variables in a problem involving Polya's Urn?

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ....
2
votes
1answer
32 views

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.
1
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1answer
34 views

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 ...
8
votes
2answers
87 views

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. ...
1
vote
1answer
71 views

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 ...
3
votes
1answer
19 views

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 ...
2
votes
1answer
58 views

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 $\newcommand{\ee}[1]{\mathbb{E}[#1]}$ $\newcommand{\pp}[1]{\mathbb{P}[#1]}$ ...