Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...
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1answer
284 views
Kullback divergence vs chi-square divergence
If the probability measures $P$ and $Q$ are mutually absolutely continuous, Kullback divergence $K(P,Q)=\int \log\left(\frac{\mathrm{d}P}{\mathrm{d}Q}\right)\mathrm{d}P$, and chi-square divergence ...
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1answer
31 views
Martingale and indicator
Exercise comes from "1000 exercices in probability" (12.9.6). Let $X_1, X_2, \dots$ be independent random variables with
$X_n=\begin{cases}
1, & \text{with probability} & (2n)^{-1}, \\
0, ...
0
votes
1answer
23 views
On unions of independent events
If the events $\{ E_{\alpha}, \alpha\in A\}$ are independent, then so are the events $\{F_\alpha,\alpha\in A\}$, where each $F_\alpha$ may be $E_\alpha$ or $E_\alpha^c$; also if $\{A_\beta, \beta\in B ...
1
vote
1answer
15 views
Calculating the probabilities of different lengths of repetitions of X length numbers
I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
0
votes
1answer
12 views
Expectation of function of stochast
I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get ...
0
votes
0answers
14 views
If the fields $F_\alpha^0$ are independent, then so are the B.F.'s $F_\alpha$.
Fields or B.F.'s $F_\alpha(\subset F)$ of any family are said to be independent iff any collection of events, one from each $F_\alpha$, forms a set of independent events. Let $F_\alpha^0$ be a field ...
0
votes
0answers
24 views
Orthogonalization of a set of random vectors
Suppose $w_1$ and $w_2$ are zero-mean jointly Gaussian random vectors. Further suppose that they have a covariance matrix given by
$$
\mathbf{cov}\begin{bmatrix}w_1 \\ w_2\end{bmatrix}
= ...
0
votes
1answer
30 views
Approximation of a random variable by a sequence of simple random variables
It said in a probability book that any non-negative random variable $X$ can be approximated by a sequence of simple random variables (finite range) $X_1,X_2,\dots,X_n$ such that ...
1
vote
2answers
60 views
Law of large numbers?
Given random variables $Z_1,Z_2,Z_3,\ldots$, which are uniformly distributed for $[8,10]$:
If $X_k =\min\{Z_1, Z_2,Z_3,\ldots,Z_k\}$, prove convergence in probability and find the constant. ...
1
vote
0answers
54 views
How does this violate probability theory?
Given: $X = Y^2 + Z^2$ (hence $E[X] = E[Y^2] + E[Z^2]$)
$p(X = 1) = .52$, $p(X = 4) = .24$, $p(X = 16) = .24$
$p(Y = -1) = .5$, $p(Y = 3) = .5$
Question: Despite not being handed any information ...
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0answers
34 views
Exponential Levy process
We assume that the stochastic process L is a Levy process with the predictable characteristics triplet $(b,c,\nu)$. Which integrability conditions we should assume for the new stochastic process
...
1
vote
1answer
16 views
Joint distribution of multiple binomial distributions
In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them.
The original file can be ...
1
vote
2answers
27 views
Uniform distribution on the n-sphere.
I have the next RV:
$$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$
where $$X_i \tilde \ N(0,1)$$
It's a random vector, and I want to show that it has a uniform ...
1
vote
0answers
15 views
Fisher information of a Binomial distribution
The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the ...
2
votes
2answers
49 views
$P[X=Y]=0$ if $X,Y$ are i.i.d. with continuous c.d.f.
I am having lots of trouble proving the following statement:
Let $X,Y$ be two real valued random variables on a probability space $(\Omega,\mathcal{F},P)$. These two variables are independent and ...
0
votes
1answer
28 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
1
vote
2answers
276 views
Finding the distribution of a function of n random, normally distributed correlated variables
Given a random vector X of n normally distributed random variables, and an n x n covariance matrix of those variables with non-zero correlation terms, what is the generalized methodology to find the ...
1
vote
2answers
72 views
Independent and uniformly distributed on $(\frac{1}{2},1]$
I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to ...
0
votes
1answer
23 views
Finding an expression for a multi variate joint CDF.
Let $X,Y$ and $Z$ be random variables with $X$ and $Y$ dependent, and $Z$ independent of both $X$ and $Y$. Let $f_{X},f_{Y},f_{Z}$ denote the density function's of $X,Y$ and $Z$ respectively and ...
0
votes
1answer
22 views
Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
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votes
1answer
65 views
Finding an expression for the probability that one random variable is less than another, given a condition.
Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as:
$$\mathbb{P}[X < Y] = ...
-1
votes
1answer
101 views
+50
How does one prove probability integral transform?
How does one prove probability integral transform? So when $Y = F_X(X)$ where $X$ has a continuous distribution for which the cumulative distribution function is $F_X$, why does $Y$ have a uniform ...
0
votes
0answers
18 views
Is there an expression for a CDF of one random variable with respect to another random variable, given a condition? [duplicate]
Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as:
$$\mathbb{P}[X < Y] = ...
2
votes
2answers
52 views
$ E\left( \left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \right) \le \left( \frac{1}{n}\sum_{j=1}^n E(|X_j|^p)^{1/p} \right)^p$
The following is problem 14 of section 3.2 from Chung's "A Course in Probability Theory".
If $p>1$, we have
$$\left| \frac{1}{n}\sum_{j=1}^n X_j \right|^{p} \le \frac{1}{n}\sum_{j=1}^n |X_j|^p$$
...
6
votes
2answers
60 views
$\lim_n \frac{1}{n} E(\max_{1\le j\le n} |X_j|) = 0$
If $\{X_n\}$ is a sequence of identically distributed r.v.'s with finite mean, then
$$\lim_n \frac{1}{n} E(\max_{1\le j\le n} |X_j|) = 0$$
The inequality
$$\frac{1}{n}E(\max_{1\le j\le n} |X_j|) ...
0
votes
1answer
144 views
Probability regarding supply and demand with Normal Distribution
The question: A sell-out crowd of $42,200$ is expected at Cleveland's Jacobs Field for next Tuesday's game against the Baltimore Orioles, the last before a long road trip. The ballpark's concession ...
4
votes
3answers
67 views
A basic doubt on Lebesgue integration
Can anyone tell me at a high level (I am not aware of measure theory much) about Lebesgue integration and why measure is needed in case of Lebesgue integration? How the measure is used to calculate ...
1
vote
1answer
39 views
Multivariate normal distribution density function
I was just reading the wikipedia article about Multivariate normal distribution: http://en.wikipedia.org/wiki/Multivariate_normal_distribution
I use a little bit different notation. If $X_1,...,X_n$ ...
1
vote
1answer
29 views
A problem on almost sure convergence
Consider a sequence of random variables defined on the standard unit interval probability space :
$ X_n = 2^n \text{when} \frac{1}{2^n} \leq \omega \leq \frac{1}{2^{n-1}}$
...
7
votes
4answers
1k views
Questions on atoms of a measure
In Kai Lai Chung's A course in
probability theory,
An atom of any probability measure $\mu$ on
$(\mathbb{R}, \mathcal{B})$ is a
singleton $\{x\}$ such that $\mu({x}) > 0$.
In Wikipedia:
...
1
vote
1answer
62 views
+50
Reverse Hölder Continuity and Hausdorff dimension
Let $f$ be a function on $[0,1]$. Say that $f$ is reverse Hölder continuous of exponent $\beta > 0$ if there is a $C >0$ such that for any $s<t\in [0,1]$, there exists $s',t'\in [s,t]$ such ...
2
votes
1answer
20 views
If $x^p P(|X|>x|)=o(1)$, then $E(|X|^{p-\epsilon})<\infty$ for $0<\epsilon<p$
If $p>0$ and $x^p P(|X|>x|)=o(1)$ as $x\to\infty$, then $E(|X|^{p-\epsilon})<\infty$ for $0<\epsilon<p$.
It feels like the assumptions should lead to something like $\sum_n^\infty ...
1
vote
2answers
45 views
Moment generating function of a stochastic integral
Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then:
$$
\mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
1
vote
1answer
326 views
Correlation between Beta distributions
I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two ...
2
votes
1answer
376 views
A consequence of the Fubini-Tonelli theorem?
Tonelli-Fubini Theorem.
Let $(\mathbb{X},\mathscr{X},\mu)$ and $(\mathbb{Y},\mathscr{Y},\nu)$ be probability spaces and let $\mathscr{Z}$ be the $\sigma$-field product i.e. the $\sigma$-field ...
0
votes
1answer
34 views
Definition of regular point of a boundary with planar brownian motion
This is an exercise in G.Lawler's book Conformally invariant processes in the plane.
First he defined regular point of a boundary using brownian motion:
Suppose $D$ is a domain in $\mathbb{C}$ with ...
0
votes
2answers
26 views
Two random variable with the same variance and mean
Let $Y\in L^{2}(\Omega,\Sigma,P)$ and let $E[Y^2|X]=X^2$ and $E[Y|X]=X$. Could we prove that $Y=X$ almost surely.
My partial answer:
By the definition of conditional expectation we have ...
3
votes
1answer
43 views
Optimal probability measure
Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
4
votes
1answer
47 views
Measurability of an Indexed Product-Measure
If for any fixed $\omega_1$, $P_{\omega_1}$ is a probability measure and $Q_{\omega_1}$ is a stochastic kernel and both are measurable in $\omega_1$, is the indexed product measure ...
1
vote
1answer
27 views
Inequality between 2p-norm and p-norm for random variables
Recently I was studying something about random matrix theory, and class of sub-guassian / sub-exponential random variables is of interest. In the literature it gave an inequality as following:
...
0
votes
1answer
60 views
probability of divisibility
Let S be the sum of k randomly selected integers between 1 and n.
What is the probability of S being divisible q?
Can this be expressed in a closed form?
This is the generalization of one of the ...
0
votes
1answer
28 views
A Measure For The Space of Probability Density Functions
Consider the space of all joint probability density functions of two variables. I want to know what the measure is of the portion of this space that is filled by uncorrelated joint pdfs relative to ...
1
vote
2answers
48 views
What exactly does this physically mean?
Let X(w) be a real random variable on ($\Omega$ , P). The image X($\Omega$) the set of all the values X(w) can take ,written $\Omega^{X}$. For any set $ B \subset \Omega^{X}$ the probability of the ...
0
votes
1answer
26 views
Convergence of random variable to a negative constant
Let $X_n$ be the sequence of R.Vs and $X_n\overset{P}{\rightarrow}A$ (or $X_n\rightarrow A$ almost surely) where $A<0$
I want to prove that $Pr[X_n < 0] \rightarrow 1$ (or $X_n < A$ almost ...
1
vote
1answer
54 views
The infinity version of Blumenthal's 0-1 law
Blumenthal's 0-1 law states that on the space of continuous maps with domain $[0, \infty)$ with the appropriate (Wiener) measure making the coordinate maps Brownian motions starting at $x$, any event ...
1
vote
0answers
24 views
Haar system and martingale
Today our lecturer used martingale theory to show that the Haar system is a basis for $L_1[0,1]$ (we're operating on the probability space $([0,1],\mathcal{B},P)$, where $P$ is the Lebesgue measure). ...
-6
votes
0answers
42 views
Borel cantelli theory [closed]
Consider the mass function $P$ defined on $\cal{F}$ by $$\displaystyle P(E_n) = \sum_{n=1}^\infty \dfrac{P(E_n)}{2^n}.$$ Show that $P(E)$ is a probability measure on $(\Omega, {\cal F}, P).$
4
votes
1answer
43 views
$P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$
If $E(X^2)=1$ and $E(|X|)\ge a >0$, then $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$.
I can see from the well known inequality $E(|X|) \le E(|X|^2)^{1/2}$ that it must be the ...
2
votes
1answer
282 views
Generalized marginal probability and marginal expectation
I think, that I need something like the following, but do not find it anywhere in textbooks. I am not even sure if it makes sense.
If you recognize it, please provvide some pointers.
Two ...
2
votes
0answers
20 views
History of odds making in sports betting
Can anyone provide a reference to the history of odds making in sports betting? In many cases, certain odds are set and then adjusted as people make bets. However, I am having difficulty tracing the ...
