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
34 views

Does weak convergence of $\nu_{n}$ imply convergence of $\int{f_{n}(x)d\nu_{n}(x)}$?

Suppose that we know that $ \int{ |f_{n}(x) - f(x)| d\mu(x)} \longrightarrow 0 \qquad (1) $ for every probability measure $\mu \in \mathcal{A}$ in a certain class. Also, suppose that $\{\nu_{n}\}$ ...
1
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1answer
64 views

Understanding PAC (probably approximately correct) bounds on the realizable case (and finite hypothesis class)

I was trying to understand PAC bounds on the realizable case (i.e. when there is some perfect $h^* \in \mathcal{H}$ and its generalization error is zero). Notation: Training data: $$S_n$$ Training ...
1
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1answer
68 views

differential equations for continiuos markov processes

I'm trying to find the forward equations for birth-and-death processes with no birth, that is, when all $\lambda$ coefficients are zero. The forward equation for a birth-and-death process has the ...
3
votes
2answers
101 views

Bounding second moment of entropy

Entropy is defined as $E(-\log(P(x))$. We know it is bounded by $\log(r)$ when $r$ is the size of alphabet. Defining the second moment as $E(\log^2(P(x))$, how to show it is bounded?
1
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1answer
58 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
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1answer
63 views

Conditioning twice?

I know that $P(X, Y)=P(X|Y)P(Y)$. How can we apply this to $P(X,Y|Z)$? We have already conditioned on $Z$, so can we condition it again on $Y$? Thanks!
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1answer
57 views

A basic question on uniform distribution [closed]

I want to know under what condition on random variable $X$, $\{\log_{10}X\}$ is uniformly distributed. Here $\{x\}$ is the fractional part of $x$.
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2answers
70 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
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0answers
49 views

Trouble in proving the simplest case of central limit theorem from a convolution viewpoint?

I have once viewed an stanford video, which proves the CLT from a convolution viewpoint rather than using the moment generating function and characteristic function etc. I felt the convolution ...
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0answers
20 views

Prove if $A_1\supset A_2,A_1; A_2\in \Im$ then $\Pr(A_1)>\Pr(A_2)$

Let $(S,\Im,P(\cdot))$ be a probability space where $\Im$ is a field denoting a collection of subsets of $S$. How can I prove that If $A_1\supset A_2, A_1,A_2\in \Im$ then \begin{equation*} ...
1
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1answer
59 views

Sequences of refining partitions of a measurable space

Let $(\Omega,\mathcal F)$ be a measurable space. For $k\in\mathbb N$ let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that each $\mathcal F_k$ is generated by a finite ...
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1answer
45 views

How can I prove that $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - (\operatorname{E}[Y])^2)/\operatorname{E}[Y^2] \:$?

How can I prove that $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - (\operatorname{E}[Y])^2)/\operatorname{E}[Y^2] \:$? I know, $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - ...
0
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1answer
70 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
2
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2answers
47 views

If $X$ is a continuous random variable uniformly distributed over $[a,b]$, then is $Y=2-4X$ uniformly distributed over $[c,d]$? Why?

I ran into this problem solving one of the problems on my course and if I knew that this applies and how to simply prove it, it would help me a great lot.
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1answer
49 views

Clique factorization

I'm reading about Clique factorization in wikipedia: http://en.wikipedia.org/wiki/Gibbs_random_field#Clique_factorization but I'm unable to understand this: What is $X_C$ here? Ok I understood ...
1
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1answer
93 views

Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...
1
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2answers
39 views

Question on Doob's martingale convergence theorem

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $(\mathcal F_k)_{k\in\mathbb N}$ a filtration of $\mathcal F$ such that $\mathcal F=\sigma(\mathcal F_k\mid k\in\mathbb N).$ Let ...
1
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1answer
49 views

$E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$

If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$ E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any ...
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0answers
87 views

Proof that a function is measurable

Suppose $f$ is a joint probability density function of random variables $X$ and $Y$. $Y$ is integrable. I need to prove that the function $g(x) = \int_{\Bbb R} f(x,y)ydy$ is measurable function. I ...
1
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1answer
86 views

Markov Chain Ergodic Theorem (Proof references)

Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff? The theorem states the following : Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively ...
1
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1answer
58 views

$X_n, n> 0$ is a Markov Chain, how to interpret $Z_n = (X_n,X_{n+1}), n > 0$?

Am a newbie to Markov Chain. So, this might be incredibly naive/stupid question. If $X_n, \, n > 0$ is MC, am having difficulty imagining/interpreting process $Z_n = (X_n,X_{n+1}), n > 0$. I ...
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1answer
152 views

finding conditional expectation under binomial distribution.

Suppose X and Y independent and are both binomial random variables with parameter N, p Compute E(X|X+Y).
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0answers
75 views

weak convergence and composition

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued ...
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0answers
64 views

Is the martingale property preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
0
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1answer
65 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
0
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1answer
101 views

Signed finite Radon measures with vague topology

If $X$ is a locally compact and $\sigma$-compact metric space. Let $M(X)$ be the space of signed finite Radon measures on $X$. (1) Show that measures with finite support is dense is $M(X)$ in the ...
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0answers
101 views

Bounds for sum of random variables

Let $A_1,...,A_M$ be random variables, not necessarily independent. For each one of them I know that $P( A_i \geq a )\leq B_i, \quad i=1,2,...,M$. How can I retrieve lower/upper bounds for ...
0
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1answer
36 views

Joint distribution and Integration

I was trying to prove a problem in my notes and now I need to whether prove or disprove the following claim: Assume $X,Y,W,Z$ are random variables defined on $(\Omega,\mathcal{F},P)$. If $(X,Y)$ and ...
0
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1answer
77 views

Showing a process is a martingale from Ito's lemma

Suppose that we have the process $t-W^2(t)$ where $W(t)$ is a Brownian motion with filtration $\mathcal{F}_t$. It is easy to show that this is a martingale by computing ...
4
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1answer
176 views

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...
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1answer
26 views

A basic question on measure

Suppose I have a measure in $B(\Bbb R)$ such that for each real number there is a neighbourhood where the measure is zero. Is that measure be necessarily zero measure ? How to prove it ? I can't take ...
3
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1answer
225 views

Why this random variable is uniformly distributed over the surface of the sphere

This is one of exercises in Probability: theory and examples, Durrett 3.2.15. Show that if $X_n = (X^1_n, ...,X^n_n) $ is uniformly distributed over the surface of the sphere of radius ...
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2answers
265 views

Probability of begin cut

Any athlete who fails the Enormous State University's women's soccer fitness test is automatically dropped from the team. Last year, Mona Header failed the test, but claimed that this was due to the ...
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2answers
41 views

question about probability problem

How is the last column calculated? I don't understand, and I don't understand the explanation. $P(A \cap B)$ is calculated by $P(A)P(B\mid A),$ right? How is $P(A\mid B)$ calculated? Thanks
2
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1answer
51 views

Finding Random variables measurable

If $[0,1]$ is our sample space and our sigma algebra is generated by all segments of the form $[0,2^{-n}]$. How can we describe the random variables measurable with respect to our sigma algebra? I'm ...
2
votes
1answer
83 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
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3answers
92 views

Solve my probability doubt? [closed]

A parent gives birth to two children. One of the child is surely a male, what is the probability of having both male child? Common answer 1/2 Actual answer 1/3
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1answer
40 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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0answers
45 views

Probability: Disease and Diagnosis

The probability of occurrence of a certain disease in a population is $1/101$. A diagnostic test has $9$ out of $10$ chances to detect the disease when the tested subject is actually affected. On the ...
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0answers
82 views

Donsker for randomly stopped processes

A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the ...
0
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1answer
36 views

Understanding Poisson Point Processes

I'm currently trying to understand PPPs. In the following I will state what I believe to know (please correct me if I'm wrong). I'm considering a PPP with intensity $\lambda$ on area $A = [-0.5, 0.5] ...
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1answer
27 views

A basic question expectation

Let $X$ and $Y$ be random variables respectively such that $E[X]=1$ and $E[Y] =0$ , $X^2 + Y^2=1, |X|\leq 1,|Y|\leq 1 $. Is it true that $X=1$ a.e. This has been used in some proof in Athreya Lahiri ...
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1answer
44 views

Inferring symmetry of a distribution from its marginals

Let $X=[X_1,\ldots,X_n]$ be a continuous random vector of size $n$ with density function $f_X(x_1,\ldots,x_n)$. If all the marginals \begin{align*} \int \ldots \int f_X(x_1,\ldots,x_n)\, ...
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0answers
220 views

Is independence a transitive property? [duplicate]

If the events $A$ and $B$ are independent and the events $B$ and $C$ are independent, does this necessarily mean events $A$ and $C$ are independent? I used coin tosses to try to model this with $A = ...
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1answer
48 views

$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

Let $Y_1, Y_2,\dots , Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
58 views

Gaussian vectors and covariance matrix.

The following is a part of a question I was given in stochastic processes course. It goes like this - I am given a series of gaussian iid random variables $\{V_i\}_{i=1}^N$ , the variable $X_0 \sim ...
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0answers
41 views

What is the relationship of the EMD (Earth movers Distance) and total variation (and other probability measures)?

I was trying to understand different methods for comparing probability distribution and saw the following paper/reference: http://arxiv.org/abs/math/0209021 In it it defines and compares and ...
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2answers
146 views

What does the sup function mean in the context of metrics for probability measures/distances/differences?

I was studying different probability metrics and distances and came across the following source: ...
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0answers
95 views

Stochastic domination by coupling

The following is a slightly streamlined version of Exercise 7.5 in Dubashi & Panconesi's "Concentration of Measure for the Analysis of Random Algorithms": Let $X$ and $Z$ be independent random ...
0
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1answer
31 views

Expected value vs values which happen with the biggest probability

If $X$ is a random variable from binomial distribution $Bin(n,p)$, then $$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$ where $p$ is the probability of one success. The expected value of random variable ...