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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Tossing a coin with at least $k$ consecutive heads

Toss a coin with $\Pr(\text{Heads})=p$ repeatedly. Let $A_k$ be the event that $k$ or more consecutive heads occurs amongst the tosses numbered $2^k,2^k+1,...,2^{k+1}-1$. Show that, $\Pr(A_k\ ...
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1answer
75 views

Is this true: probability independent from i?

We have a set of i.i.d. random variable $X_i$ with some discrete distribution. Further we have a random variable Y, Independent from $X_i$ with a Binomial Distribution Bin(n,p). Now we are ...
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34 views

Computing equilibrium measure for Borel sets eg. Ball

I am asking for methods to compute equilibrium measures. The more the better. Here is the definition of equilibrium measure in the Brownian motion setting: Let $\gamma=\sup\{t\in [0,T]: B_{t}\in ...
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54 views

Sums of Power Law random variables

Suppose $F$ be a pareto distribution with scale parameter $x_m$ and shape parameter $\alpha$. Assume $X_1, X_2 , ..., X_n$ are iid random variables drawn from $F$. Let $S_n(k) = X_1 ^k + X_2 ^k + ...
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54 views

A simple (?) equality of two initial $\sigma$-algebras

I'm afraid I miss the forest for the trees.... Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(E,\mathcal{B}_E)$ be a state space and $\mathcal{E}$ a generating $\pi$-system of ...
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55 views

What is a.e. a.s

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such ...
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19 views

A question in a textbook about Blumenthal 0-1 law for a general Markov process

This question came up as a result of reading this question . Here is the Blumenthal 0-1 law in the book Stochastic Processes by Richard F. Bass. Proposition 20.8 Let $(X_t , \Bbb{P}^x)$ be a ...
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1answer
114 views

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ...
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1answer
241 views

Precise definition of the support of a random variable

I am reading lecture notes which contradict my understanding of random variables. Suppose we have a probability space $(\Omega, \mathcal{F}, Pr)$, where $\Omega$ is the set of outcomes ...
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24 views

Multiple events happening almost surely

I am reading an article about reinforced random walks from B.Davis. In the summary the author basically says that there are two events , I call it $A$ and $B$, and that either $A$ happens almost ...
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1answer
60 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle ...
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1answer
29 views

$L_p$ spaces and tail estimates

I can prove the main identity in this question. Not sure how the "and deduce" bit works. I think $O(\lambda^{-q})$ is some kind of tail estimate.
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1answer
27 views

Probability “average” understanding

This is more of a problem understanding probabilities than an actual question. In a game I am playing I can use a certain item to try to unlock different levels. The item will unlock a new level ...
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2answers
71 views

Show independence of $(aX,bX^2)$, $\,X\sim N(0,\sigma^2)$?

How can we proof that $aX,bX^2$ are independent iff $b\cdot a=0$, when $\,X\sim N(0,\sigma^2)$? I found that $X^2$ is Chi-Square distributed, and the correlation is: ...
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1answer
25 views

Modes of Convergence of a particular random variable

Let $X_n \sim U([-1/n,1/n])$ be uniform random variables on $[-1/n,1/n]$ for $n \in \mathbb{N}$. Do the $X_n$ converge, and if yes in what sense? I think it converges pointwise as for any $x \in ...
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14 views

Matrix Completion feature vectors

In the noisy matrix completion problem, with enough number of revealed entries say $|E| = nr^2 log(n) $, can we have a bound on the error in the singular vectors of a sub matrix. For, example say ...
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2answers
46 views

evaluating an integral with complex exponential (spectral density)

I am having a hard time figuring out how to evaluate this integral from a book that I am reading. Here's the background info but I doubt it's highly relevant to the problem at hand: $X$ is a real ...
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50 views

Why is the Stopping Theorem interesting?

The theorem for discrete-time martingales is as follows: Let $X=(\Omega,\mathcal{F},(\mathcal{F}_n)_n,(X_n)_n,\mathrm{P})$ be a supermartingale and $\tau_1,\tau_2$ two a.s. bounded stopping times on ...
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1answer
60 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
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46 views

Why does a Gaussian process have a gradient whose determinant is Gaussian?

I'm trying to understand something in Adler and Taylor's book, Random Fields and Geometry. Let $T \subset \mathbb{R}^N$ be a compact parameter set (for simplicity, suppose it is a closed hypercube) ...
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2answers
26 views

Inverse function of borel sets when function is a constant.

Following a simple proof my professor explained in class I am having problems with a specific step: The proof is of probabilistic nature and we are trying to prove that If $X$ (random variable) is ...
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49 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
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1answer
59 views

Measures in conditional expectation.

I always make confusion when a measure has to be changed in some other measure. This time I'm stuck on a change of measure in the definition of conditional expectation of a random variable. If $Z$ is ...
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1answer
65 views

Characteristic function of logarithm of random variable

If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$? I know that $\phi_X(t)=E[e^{itX}]$ and ...
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29 views

The distribution of ratio of two shifted gamma

I am wondering if anyone can help me to find the ratio of this distribution. Assume $S$ and $T$ are independent, where $S\sim Gamma(n-1/2, 4(1+\rho)\sigma^2)$ $S\sim Gamma(n-1/2, ...
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3answers
82 views

joint probability equals to marginal probability

Is there any situation in that joint probability $p(x,y)$ equals to marginal probability $p(x)$? what is the interpretation of this situation?
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1answer
133 views

Square root of Chi-square distribution tends to $N(0,1)$

The question requires to show that $\sqrt{2\chi^2_n}-\sqrt{2n}$ converges in distribution to $N(0,1)$ as $n \rightarrow \infty$, which I dont know how to proceed. The question also has a first part ...
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1answer
47 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
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1answer
77 views

An exponential martingale [closed]

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
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36 views

If $\{X_n\}$ is a Cauchy seq of r.vs a.s., then why does it converge to some r.v?

I was suddenly suspicious of what title says. My logics are follows. $\{X_n\}$ is a Cauchy seq of r.vs $P$-a.s. $\Leftrightarrow$ $P(\{X_n\}$is a Cauchy $)=1$ $\Leftrightarrow$ $\exists ...
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1answer
105 views

Proving it's a martingale and more conditions.

Let $(X_{n})_{n>0}$ be a sequence of random variables in $[0, 1]$ and assuming that ($X_{0}=a) \epsilon [0, 1]$ then: $Pr\left(X_{n+1}=\frac{X_{n}}{2}|\mathcal{F}_{n}\right)=1-X_{n}$ and ...
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0answers
35 views

continuous, non-additive set function

I am looking for a non-additive, continuous set function from a simplex $\Pi_{n}$ of finite dimension $n-1$ into $[0,\infty]$. The motivation is as follows. Shannon defined the entropy ...
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103 views

The completeness assumption in Prokhorov's theorem

Originally, I encountered this question on Terence Tao's blog, where the following exercise is presented: Exercise 23 (Implications and equivalences) Let $X_n, X$ be random variables taking values ...
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43 views

How to show $ P\big(\big|\frac{X}{n}-p\big|>a\big)\le\frac{\sqrt{p(1-p)}}{a^2n}min\big\{\sqrt{p(1-p)},a\sqrt{n}\big\}$

Let $X$ be binomial, $B(p,n)$ with $p>0$ fixed, and $a>0$. Show that, $\displaystyle ...
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1answer
94 views

What are the implications of the definition of limiting distribution?

Given a sequence of random variables $\{X_n\}$, if $$\lim_{n\to\infty}F_{X_n}(x)=F_X(x),\qquad\forall x\in C(F_X),$$ then we say $X$ is the limiting distribution of $\{X_n\}$. My question is: Under ...
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0answers
32 views

Random 0-1 matrices [duplicate]

For $n,r∈N$, $1<r<n$, let $z(r,n)$ be the largest possible number of 0 entries in an $n×n$ matrix which has no $r×r$ submatrix whose entries are all $0$. (Here a submatrix is obtained by ...
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100 views

Random $0-1$ matrices

I'm working my way through the Oxford notes in Probabilistic Combinatorics and came across this question in one of the question sheets; I'd like to stress that this is not my homework: I'm simply ...
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43 views

Approximate the distribution of the sum of ind. Beta r.v

If $X_i$ has a Beta distribution $\beta(1,K)$. What is the best approximation for the distribution of $ S=\sum_{i=1}^N X_i$, when the $X_{i}$ are independent and $N$ is finite. Thanks
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1answer
55 views

Question about almost sure convergence.

I am struggling a little to understand almost sure convergence in probability theory. I have taken some general measure theory and there we had abot convergence almost everywhere. Basically it was ...
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1answer
189 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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3answers
106 views

Expected value of the function of a random variable

I am studying Probability and Monte Carlo methods, and it feels that the more I study the less I truly understand the theory. I guess I just confuse myself now. So the expected value of a random ...
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1answer
32 views

Showing independence of rectangular events…

Suppose I have a sequence of independent random variables $\{X_n, n \in \mathbb N\}$. How do I show formally that $P((X_1,...,X_n)\in A, (X_{n+1},...)\in B) = P((X_1,...,X_n)\in A)P((X_{n+1},...)\in ...
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9 views

Limiting expression for Power law tail index from a quantile function?

Assume that I have a random variable $X$ (which I know will have a power law tail). If I had the CDF for $X$, $G(x)$, then I could easy calculate this tail as something like, $$ \alpha = ...
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1answer
30 views

find the marginal probabilty density function

suppose that X and Y are continous random variable with joint pdf given as follows : f(x,y)= { 15y for x^2 <= y <= x } find the marginal probabilty density function?? how can i find marginal of ...
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1answer
49 views

Convergence almost surely and B-C lemmas

Showing the expectation is straightforward. I am not sure how to use the Borel-Cantelli lemmas to show the almost surely part.
2
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1answer
53 views

Convex Hull of discrete points

If i was to give an $n \times n$ grid with each grid point having probability $p$ of being selected, would it be difficult to calculate distributions of various measures regarding the convex hull of ...
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2answers
86 views

Are there any cases where $\mathbb E(|X|)<\infty$ and $\mathbb E(X)<\infty$ aren't equivalent?

I often see $\mathbb E(|X|)<\infty$ among the givens in a statement. That made me wonder: why not just demand $\mathbb E(X)<\infty$? In the light of the theorem Let $f$ be measurable. Then ...
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1answer
271 views

Proof of Pearson's chi squared test

i was reading proof of this theorem on http://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec23.pdf They showed, that $\frac{v_j-np_j}{\sqrt{np_j}} ...
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3answers
255 views

What is a fair game?

Suppose $X_n$ is the fortune of a gambler after $n$ th game. Then the game is called fair (Breiman 1968) if $$E[X_{n+1} \mid X_1, \dots, X_n] = X_n \forall n$$ My question is why a fair game is not ...
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1answer
50 views

Probability of the same position!

If we have $n$ different boxes in a room, each with $x$ slots (positions $1,2,\ldots,x$) each and we have $(x-1)$ red balls and $1$ white ball in each box on different slots. What is the probability ...