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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1answer
87 views

What's the distribution function

I have a random variable $1_{\{az_1+bz_i<L\}}$, where $1_{\{.\}}$ is the indicator function, $z_1,z_i$ are $N(0,1)$, i.e., i.i.d standard normal and $L$ is a constant. If $X = ...
4
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1answer
109 views

Borel functions and their equivalence to sets.

Let $(\Omega,\mathcal{F})$ be a measurable space. The following are equivalent: $\ X:\Omega \to \mathbb{R} $ is a Borel function. $\{\omega\in\Omega:X(\omega)>a\}\in\mathcal{F}$ for all ...
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0answers
122 views

Uniform Integrability on Compact sets

Let $m$ be a probability measure on the compact set $W \subset \mathbb{R}^m$, so that $m(W)=1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, ...
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1answer
138 views

Why is $X_1 + X_2 +\ldots + X_n$ a martingale?

If we have $X_k$ random variables with average $0$ and independent, why is the $\sum_{k=1}^n X_k$ a martingale for the sigma algebra $\mathcal F_n$ generated by $\{X_1,\ldots, X_n\}$? I basically ...
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1answer
98 views

Random variables X, G are have functional relationship G=g(X). How does g relate the graphs of their distributions?

More specifically, as an educational tool I want to prepare a slideshow showing (in 2-D) the graph of $F_X$ transforming into the graph of $F_G$. (I think it can be done in 4 steps (e.g., graphs on ...
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1answer
93 views

Expression for $n$-th moment

I stumbled upon an expression in an article of statistics for an $n$-th moment with $X$ being a random variable over $[0, \infty)$. $$\mathbb{E} X^{n} = \int^{\infty}_{0} nz^{n-1}\; \text{Pr}(X > ...
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2answers
230 views

Measurable funtion and sub sigma algebra

If $X_t$ is measurable w.r.t a sigma algebra $F_t$, what is the relationship between $X_t$ and a sub-sigma-algebra $F_{t_1}$ s.t. $F_{t_1}$ is included in $F_t$ ? A preliminary thought question, ...
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2answers
433 views

Random variable with mean $\mu$ and variance $\sigma ^2$

I have never taken probability theory, and I wonder whether one can express some random variable $X$ with mean $\mu$ and variance $\sigma ^2$ in terms of $\mu$ and $\sigma$ only. Or at least ...
4
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1answer
246 views

Is the set of all probability measures weak*-closed?

Let $(\Omega,\Sigma)$ be a measurable space. Denote by $ba(\Sigma)$ the set of all bounded and finitely additive measures on $(\Omega,\Sigma)$ (see http://en.wikipedia.org/wiki/Ba_space for a ...
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1answer
233 views

When weak convergence implies moment convergence?

Given a sequence $(\mu_n)_n$ of probability measures on $\mathbb R$, which converges weakly to a probability measure $\mu$, when do we have $$ \lim_{n}\int x^kd\mu_n(x)=\int x^k d\mu(x) \qquad ...
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1answer
197 views

Quick question about lim and sup

I've got a question regarding a step in a proof, the situation is following: Let $X_{1},\dots,X_{n}$ be independent, symmetric stochastic variables so that $\sum\limits_{n=1}^{\infty}X_{n}$ exists in ...
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0answers
71 views

Question (3) on Uniform Integrability (simpler)

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W)=1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, $X \subseteq \mathbb{R}^n$ such that $\forall w \in W$ $\ ...
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1answer
158 views

Partial fractions for geometric probability-generating function wrong

Let $X\sim \text{Geo}(1/4), Y\sim \text{Geo}(1/2)$ be given. First I have to compute $\mathbb{E}[z^{X+Y}]$: ...
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3answers
565 views

Is the support of a random variable those values where the graph of its distribution is not “flat”?

In the literature the support, $S$, of a random variable $X$ is defined as the smallest closed subset of real line $\mathbb{R}$ with probability $1$. Looking to prove that $S$ is where the graph of ...
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2answers
240 views

Simplification of the Expected Value via CDF: Does it work for ALL Probability Distributions?

If a random variable $X$ has a density $f$, then the expected value can be simplified: $$\mathbb{E}[X]=a+∫_{a}^{b}(1-F(x))dx,$$ where $F$ is the cumulative distribution function, $F(x)=\Pr(X≤x)$. ...
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0answers
319 views

Topological necessary and sufficient condition for tightness

Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$: For each $\varepsilon>0$, we can find a compact subset $K$ of ...
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0answers
235 views

Question (2) on Uniform Integrability

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W) = 1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, such that ...
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2answers
4k views

Basic probability - either event occurs but not both

I'm taking a graduate course in probability and statistics using Larsen and Marx, 4th edition and I'm struggling with a seemingly basic question. If A and B are any two events, not mutually ...
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1answer
91 views

equivalence of $E[X_\infty]=1$ and $X$ is a u.i. martingale on $[0,\infty]$

Let $(X_t)$ be a strictly positive supermartingale on $[0,\infty)$. Hence $X_t$ covnerge to $X_\infty$ a.s. Now how can I show the following: $E[X_\infty]=1$ is equivalent to $(X_t)$ is a uniformly ...
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1answer
185 views

Maximally entropy preserving irreversible functions. (CS related)

The topic/problem is related to hashing for data structures used in programming, but I seek formal treatment. I hope that by studying the problem I will be enlightened of the fundamental limitations ...
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1answer
410 views

A generalized Jensen's inequality

Let $(U , \mu)$ and $(V,\nu)$ be probability spaces. Let $f$ be a convex functional on $L^1(\mu)$, i.e. $$f(tX + (1-t)Y) \leq t f(X) + (1-t)f(Y)$$ for all random variables $X$ and $Y$ in $L^1(\mu)$. ...
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2answers
173 views

a question on general conditional probability

Let $(\Omega ,\mathcal F ,P) := \bigl((0,1],\mathcal B((0,1]),u \bigr)$, where $u$ is the Lebesgue measure restricted to $\mathcal B((0 ,1])$. Let $X\colon\Omega\to\mathbb R$ be defined by $X(\omega) ...
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7answers
3k views

Good books on “advanced” probabilities

what are some good books on probabilities and measure theory? I already know basic probabalities, but I'm interested in sigma-algrebas, filtrations, stopping times etc, with possibly examples of ...
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1answer
5k views

expectation of product of independent random variable

it is a problem from my exam praperation sheet Let $U$ , $Y$ be independent random variables. Here $U$ is uniformly distributed on $(0 , 1)$ . whereas $Y$ is $\frac{1}{4} \delta(0) + \frac{3}{4} ...
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1answer
566 views

How to use joint characteristic function to calculate characteristic function for single variables? [duplicate]

Possible Duplicate: probability question on characteristic function It is a problem in my practice exam. Defined on some common probability space, two random variables $X$, $Y$ have the ...
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1answer
105 views

Deriving the characteristic function for $N(0,2)$

Could someone please help me with an easy derivation of the characteristic function for a $N(0,2)$ distribution? Or a link to somewhere it is done.
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1answer
412 views

probability question on characteristic function

I got a big problem with my exam practice question on characteristic function. Here are two. Let $X$, $Y$ be two independent random variables with the following characteristic functions: ...
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0answers
122 views

Distributional Derivative For Stieltjes Integration

For the sake of example, suppose we are on the real line and that $H(x)$ is the Heaviside step function and $\delta(x)$ is the Dirac Delta function. According to the theory of distributional ...
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1answer
86 views

Question on uniform intergrability

Consider a probability measure $m$ over $W \subseteq{R^m}$, so that $m(W) = 1$. Consider a function $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, with compact $X \subset \mathbb{R}^n$, such that ...
0
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1answer
159 views

Measure-theoretic view of expectation of a Bernoulli sequence

Problem: I have a good understanding of basic Bernoulli and Binomial RVs, but this was foundational work in statistics. I am attempting to try and apply my (minimal but increasing) knowledge of ...
3
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1answer
132 views

Form of $\sigma(X_n)$ and $\sigma(X_n)$-measurability

Problem: Suppose $\tilde{X}=(X_1,X_2,...)$ is a sequence of RVs on $(\Omega,\mathcal{B})$. Prove that $\sigma(\tilde{X})$ is generated by events of the form: $\bigcap_{i=1}^m \{X_i\leq x_i\}$ for ...
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3answers
2k views

Coin Toss Probability Question (Feller)

I'm working out of Feller's "Introduction to Probability and its Application (Vol I.)" textbook and I'm stuck on a coin toss problem. I'll list the full problem and show where I'm having trouble. ...
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1answer
386 views

Integrals and suprema

If I know that$\ X < \sup_iX_i$ for example and if $\ X_i $ and $X$ are integrable then when I integrate both sides, where do I put the integral on the RHS? I know that in some situations you can ...
2
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1answer
133 views

Weak a.s. convergence VS a.s.weak convergence

Let's consider a sequence $(\mu_n)_n$ of random probability measures on $\mathbb R$, and let $C_b$ be the Banach space of bounded continuous functions on $\mathbb R$. I am considering the following ...
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1answer
138 views

Convergence a.s , Convergence in probability and convergence in mean

Can anybody help me, Im trying to think of an example to show that convergence in mean does not imply convergence P a.s and also to show in the other direction, i.e convergence in P a.s does not imply ...
2
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1answer
133 views

Uniform Integrability after composition

Let $\mu$ be a finite measure on $X \subseteq \mathbb{R}^n$. Consider the Uniformly Integrable family $\{ f_n(\cdot) \}_{n \in \mathbb{N}}$ of functions $f_n : X \rightarrow \mathbb{R}_{\geq 0}$. ...
0
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1answer
240 views

Conditional Expectation.

Are the following two the same: $E[V(X_{t_{k+1}})|g(X_{t_{k+1}}),X_{t_k}]$ and $E[E[V(X_{t_{k+1}})|g(X_{t_{k+1}})]|X_{t_k}]$ Where $X$ is Markov chain $X_{t_k} \in \mathcal{R}^n$ $V: ...
2
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0answers
280 views

Measurable indicator function that cannot be approximated by a continuous sequence $f_n$

Resnick's exercise 3.15 asks the following: Suppose $-\infty<a\leq b<\infty$. Show that $1_{(a,b]}(x)$ can be approximated by bounded and continuous functions. That is, show $0\leq f_n \leq 1$ ...
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6answers
504 views

Variation of the Monty Hall Problem.

Suppose instead of the normal Monty Hall scenario in which we have two empty doors and a car residing behind the third, we instead have three types of objects. One is a car, one is a hard drive, and ...
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0answers
202 views

Determining a square integrable martingale

I'm preparing for an exam in my course Martingales & Stochastic Integrals. Currently I'm having a look at some old exams, and there's a question on one of them that I'm not able to figure out. The ...
2
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1answer
158 views

Is the expectation $E[\xi U'(\xi)]$ finite?

I encounter the following problem today. It seems a simple question. Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions: (1) $U$ is concave, continuous, ...
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2answers
2k views

Ratio Distribution: Poisson Random Variables

Suppose two Poisson processes. For example, during the time interval, $\Delta t_{1} = t_{1} - t_{o} = 50\mu s$ , $x$ photons are incident on a detector with rate $\lambda_{1} = 10$x$10^4 s^{-1}$. At ...
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2answers
1k views

How to transform normally distributed random sequence N(0,1) to uniformly distributed U(0,1)?

Everybody knows how to convert U(0,1) to N(0,1). However does anybody know an efficient algorithm solving the opposite task? I mean how to generate U(0,1) sequence from N(0,1) one? Asking because a ...
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0answers
54 views

Algebra involved in computations with (or extensions of) a probability measure on a lattice

Suppose we have a probability measure $\gamma$ defined on the $d$-dimensional lattice $\mathbb{N}^d$. Let us say $d = 5$ for simplicity. I will also write $\gamma_{1:5}$ for this measure and write say ...
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1answer
284 views

Explicit examples of smooth entropy computation

Smooth classic entropies generalize the standard notions of entropy. This smoothing stands for a minimization/maximization over all events $\Omega$ such that $p(\Omega)\geq 1-\varepsilon$ for a given ...
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1answer
3k views

Tower property of conditional expectation

I'm trying to prove the "tower property" of conditional expectations, $$ E[V\mid W] = E[\ E[V\mid U,W]\ \mid W\ ], $$ where $U$, $V$ and $W$ are any random variables. $E[X \mid Y]$ is itself a ...
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0answers
64 views

uniform integrability allows to apply limits

If I have the following expression: $$E[M_{t\wedge n}|\mathcal{F}_s]$$ Where the set of random variables $M_{t\wedge n}$ is bounded in $L^2$, i.e. $$\sup_{t} E[M^2_{t\wedge n}]<\infty$$ Hence ...
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1answer
176 views

Almost sure convergence for sequence of function

Suppose you have 2 probability spaces $(\Omega, \mu)$ and $(\Psi,\lambda)$. For every $t\in (0,1)$ let $f_t$ be a real-valued non-negative measurable function bounded by one, that is $f_t: ...
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2answers
2k views

Does convergence in distribution implies convergence of expectation?

If we have a sequence of random variables $X_1,X_2,\ldots,X_n$ converges in distribution to $X$, i.e. $X_n \rightarrow_d X$, then is $$ \lim_{n \to \infty} E(X_n) = E(X) $$ correct? I know that ...
2
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0answers
253 views

Doob inequality for continuous martingales

In our class we have proven Doob's inequality for discrete martingale, i.e. Let $(M_n)_{n \in \mathbb{N}}$ a martingale, then $$ \| \max_{0\le k\le n} M_k\|_p \le C_p \|X_n\|_p$$ for $p\in ...