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 ...

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Continuity of $\mu \mapsto \mu(E)$ for $\mu$ probability measure and $E$ Borel subset

Let $X$ be a topological space endowed with the Borel sigma-algebra, let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, endowed with the weak* topology. Fix $E$ Borel subset of ...
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1answer
4 views

If Y and X are ind. binomial RV with parameters (n, p) and (a, b) respectively, then (Y/n) - (X/a) is approximately distributed. Find V(Y/n - X/a).

I tried to find E(Y/n - X/a) and said it was E(Y/n) - E(X/a)= p - b. But then I got stuck finding the variance, I wasn't sure if it needed to be done with moment generating functions or if the Central ...
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1answer
13 views

Show that if the sum of an diverges, no discrete probability space can contain independent events

Suppose that 0$\leq$ $p_n$ $\leq$ 1 , and put $a_n$= min {$p_n$, $1-p_n$}. Show that if $\sum a_n$ diverges, then no discrete probability space can contain independent events $A_1$, $A_2$, .... such ...
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0answers
12 views

Sample Space of a set

Let $S = \{1, 2, \cdots , n\}$, and $T = \emptyset$. Repeat the following step as long as $|T| \lt n$: Uniformly at random pick a number $n$ from $S$; if $n \not \in T$, then $T = T \cup \{n\}$. As ...
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1answer
21 views

Find three events that are dependent but pairwise independent

http://imgur.com/yMh3D0o really bad at formatting... so My logic on this is something like a spinner with equal chances for all 4 and then if you get a certain number you flip a coin or something ...
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30 views

Prove that the following is a field.

http://imgur.com/xMsuhBf 2.4a Honestly, I'm really confused with this notation and what a field is etc. I get the 3 conditions for it to be a field I'm just really bad at proving them. So far I ...
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1answer
16 views

Does this argument suffice to show a “record” occurs at time n with probability 1/n?

I think it does, but, in addition to checking for correctness, I'd like to know what other argument we might use. Let $X_1, X_2,...X_n$ be be a sequence of independent identically distributed ...
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0answers
21 views

Prove about weak convergence

Let $X_n\Rightarrow X$ and $Y_n\Rightarrow c$. Show what $X_n+Y_n\Rightarrow X+c$. Prove: There exists sequences of random variables $(X^{(*)}_n)$ and $(Y^{(*)}_n)$ such that $(X^{(*)}_n)$ and ...
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0answers
21 views

Bayes Theorem and Probabilities [on hold]

Suppose that economic outcomes can be classified as either good or bad. Governments differ in ability and this affects the likelihood of good outcomes. There are two types of governments: high ability ...
2
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1answer
14 views

Fatou for weak convergence

I want to do exercise 3.2.4 from Rick Durett, Probability: Theory and Examples page 86. $$\text{Let } g\geq0 \text{ be continuous. If }X_n \Rightarrow X_{\infty} \text{ then } \liminf_{n\rightarrow ...
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0answers
8 views

Expected first return time of Markov Chain

Given the following Markov Chain: $$M = \left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{4} & \frac{3}{4} & 0 & 0 & 0 & 0 ...
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1answer
19 views

Conditional probability density function

Let $\theta$ be the parameter of the probability density function $f(x)$. If it is mentioned that $f(x|\theta)$ be the conditional probability density function, then what does $f(x|\theta)$ mean? ...
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1answer
25 views

Writing random variable formulas with set notations, What is the problem?

Is it wrong to write $\displaystyle P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$ when $X$ and $Y$ are random variables? As I know a random variable is a function and therefore has a range and the two ...
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1answer
20 views

Prove that if $E(X\log X)<\infty$ then $E(\sup_n |S_n|/n)<\infty$.

This is part 2 of a two part question. In the first part, we were asked to show that if you had a non-negative sub martingale $M_n$ then $$\sup_n E(\sup_{k\leq n} M_k)\leq \sup_n 2E(M_n \log M_n)+2$$ ...
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2answers
45 views

Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
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12 views

Calculate Expectation for bin related problems

Suppose that there are n balls numbered 1 through n and there are n bins numbered 1 through n.I'm finding difficult to understand on how to calculate Expectation X which is number of bins with exactly ...
2
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1answer
15 views

Conditional expectation of $Y_1$ given that $\sup Y_i=z$, for $(Y_i)$ i.i.d. uniform on $[0,\theta]$

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
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0answers
9 views

Rao-Blackwell improvement for a nonrandomized estimator

Context: please consider a parametric statistical model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ and suppose that we are estimating $g(\theta)$. Associated with this is the set of decisions ...
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2answers
43 views

Random sums of iid Uniform random variables

Let $\{X_r : r\ge 1\}$ be independently and uniformly distributed on $[0,1]$. Let $0<x<1$ and define $$N=\min\{n\ge 1 : X_1 + X_2 +\ldots+X_n> x\}$$ Show that $$P(N>n) = ...
2
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1answer
19 views

Equivalent condition for $ (X_n, \mathcal{F}_n) $ to be a martingale

I've encountered an interesting problem and am not quite able to solve it. It is to prove the following statement ($ X_n $ denotes a sequence adapted to a filtration $\mathcal{F}_n) $: $$ ...
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0answers
15 views

Why is “having countably many open rays” a measurable condition

In discussing Bernoulli($p$) percolation on a tree, sometimes one asks the question of what the probability is that there are countably infinitely many rays containing only open edges. I don't see ...
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15 views

a way to compute energy of a flow on transient trees

Let $T$ be a rooted tree that is also a transient electrical network (so effective resistance from root to infinity is finite) but with recurrent rays. (So effective resistance root to infinity along ...
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0answers
22 views

Repeated coin flips probability [on hold]

Assume in an experiment, one flips a coin $L$ times. This experiment is repeated T times. Assume the $k$'th flip for all possible $k$ values ($1 \le k \le L$) among all experiements. If the head ...
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0answers
33 views

Can anything be learned about a probability distribution *directly* from its characteristic function?

Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us ...
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1answer
42 views

What does it mean that an expected value does not exist?

$X$ is a random variable with pdf $f$ and $g: \mathbb R \to \mathbb R$ is a measurable function. Before I start operating with $E[g(X)]$ I need to show that it exists. What does it take to show it? ...
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0answers
20 views

Deducing results about continuous time random walks from the corresponding discrete time result

Is there any standard way to prove results about continuous time random walks from the corresponding results for discrete time random walks? Specifically, my problem is that I was reading Lawler and ...
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0answers
6 views

M/M/1 Queue Under Heavy/Light Loads

Given, $\rho \approx 1$ (heavy-load), for the open M/M/1 queue, can we consider the sequence of response times an approximate renewal process? That is, the sequence of measured response times from a ...
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2answers
74 views

Strange behaviors of finitely additive probabilities

Watching a lecture on youtube I heard the lecturer stating that in general finitely additive probabilities behaves strangely. For example, it is possible that every open interval around a point $x$ ...
3
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1answer
41 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
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27 views

conditional probability proof 3 varables [on hold]

Suppose that $\mathcal a$ ,$\mathcal b$ and $\mathcal c$ are dependent variables. $$\mathbb P(a \mid b) = \sum \mathbb P(a \mid b,c) \ \mathbb P(c \mid b)$$ can anyone explain it how we get it?
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1answer
34 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
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1answer
41 views

Give simpler examples of $L^p$ unbounded yet converging martingales

I read the following example in the book Counterexamples in Probability and Real Analysis by Gary L. Wise and Eric B. Hall: Does anyone know simpler examples? I do have one! I would be glad to ...
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1answer
16 views

Question about “integrable” random variable

I was reading the definition of Markov's Inequality on Wikipedia and it says If $X$ is any nonnegative integrable random variable and $a > 0$, then $\mathbb{P}(X \geq a) \leq ...
2
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2answers
28 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
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1answer
27 views

Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
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0answers
32 views

ODE for the normal distribution [on hold]

The normal density function $\phi(x)=\tfrac{1}{2\pi}e^{-\frac{x^2}2}$ can be described via the ODE $$\phi^\prime(x) = -x \phi(x)$$ under the condition $\int_{-\infty}^\infty \phi(x) = 1$. Is there ...
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0answers
28 views

How to demonstrate the pdf of $P_{\sigma} (t)=\lambda_c e^{- \lambda_c t} / (1 - e^{- \lambda_c T})$

In $t_c$, there are $n$ expirations of $T$ and the remnant $\sigma$ seen from the above figure. Let the time $t_c$ forms the exponential distribution with parameter $\lambda_c$. How to demonstrate ...
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20 views

Dominated Convergence Theorem.

Dominated Convergence Theorem "Suppose $X_{n}\rightarrow X$ a.s., and there is a random variable $Y$ with $E[Y]<\infty$ such that $|X_{n}|<Y$ for all $n$. Then $E[lim_{n \to ...
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1answer
35 views

A Seemingly Trivial but Computationally Complicated Probability Problem

Suppose $X,Y$ are independent $Uniform(-1,1)$ random variables. Determine the distribution of $Z=X-Y$. I do not really think I should add my work here because whatever I have tried until now, has ...
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0answers
21 views

Limit definition of Sets.

Proposition 1.32 $X_{n}\xrightarrow{a.s.} X$ if and only if for any $\epsilon>0$ $P( | X_{n}- X |<\epsilon, \; \forall n\geq m )\rightarrow1$ $as$ $ m\rightarrow\infty$ Proof. Suppose first ...
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1answer
22 views

ways to choose 4 people including two or more male from N people with 1/3 male 2/3 female

A sample of four people is randomly drawn from a population of N > 4 people.Assume that 1/3 of the total population is male, and 2/3 is female. (To simplify things, let’s assume that N is always ...
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0answers
25 views

How to prove stochastic dominance? [on hold]

Consider the set of constant vectors $p_i$ and $\tilde{p}_i$, such that $p_i \succeq \tilde{p}_i \succ \mathbb{0}\; \forall i$ (component wise inequality) and define: $M \triangleq ...
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1answer
25 views

Doob-Kolmogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
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1answer
15 views

Determining density involving scaled beta distribution

Suppose $Y \sim \mathrm{Beta}(2,1)$. If $X = \theta{Y}$ (for some $\theta > 0$) how do I determine the joint density $f(x, \theta)$? Edit: the density for $Z$ is $2z$. Would it be correct to say, ...
2
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1answer
16 views

Showing that the union of two algebras will give us a disjoint finite union of sets.

Suppose we have a space $X$ that could be anything and $\mathcal{A}_{1}$, $\mathcal{A}_{2}$ are algebras. How would I show that that algebra generated by $\mathcal{A}_{1} \cup \mathcal{A}_{2}$ ...
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1answer
28 views

Law of total probability explanation

What is the intuition behind the law of total probability? http://en.m.wikipedia.org/wiki/Law_of_total_probability
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1answer
25 views

Independent events and Kolmogorov

Suppose we have a probability space $(\Omega, \mathfrak{F}, P)$, and independent events $(E_n)_n$. Consider $$M_n = \sum_{k=1}^n I_{E_k}$$ Is it correct to say that by the Kolmogorov $01$ law $M_n$ ...
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2answers
25 views

Need a Probability Theory book that also focusses on Analysis

I am in search for a Probability Theory book which also contains elements and proofs from Analysis. A non-Measure Theoretic approach is most desirable. I have gone through great books like Ross but I ...
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8 views

Approximation of objective based on statistical distance

I am a computer science researcher (mostly theoretical) currently in midst of statistics and not able to figure out how to proceed. At an abstract level, I have a hypothesis for an unknown ...
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28 views

Poisson Process Alterations

If we have a Poisson process of rate $\lambda$, do the following alterations still result in a Poisson process? 1) Deleting every alternate point 2) Inserting points at times $1, 2, 3, ...$ 3) ...