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 ...
3
votes
1answer
64 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 ...
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:
...
1
vote
2answers
52 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 ...
0
votes
1answer
30 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
49 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 ...
4
votes
1answer
51 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 ...
0
votes
1answer
61 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 ...
1
vote
0answers
25 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). ...
4
votes
1answer
44 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
0answers
23 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 ...
0
votes
1answer
21 views
Conditional expectation is square-integrable
I am given the following definition:
Let $(G_i:i\in I )$ be a countable family of disjoint events, whose union is the probability space $\Omega$. Let the $\sigma$-algebra generated by these events ...
3
votes
4answers
98 views
Probability of sequence
The question is as follow: Let $(X_{n})_{n}$ be a sequence of Random variable that is independent with the probability
$P(X_{n}=1)=1-P(X_{n}=0)=\frac{1}{n}$
Show that $P\left( \underset { ...
-6
votes
0answers
43 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}.$$ Show that $P(E)$ is a probability measure on $(\Omega, {\cal F}, P).$
5
votes
1answer
56 views
Applications of information geometry to the natural sciences
I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
2
votes
1answer
45 views
An application of Donsker's theorem.
Let ${X_i}$ be iid with $E[X]=0$ and $Var(X)=\sigma^2$ Let $S_0=0$ and $S_n=X_1+...+X_n$ for all $n \ge 1$. Show that $\lim_{n\rightarrow \infty}P(S_k>0 \space for \space k=n,n+1,\dots,2n)=1/4 $.
...
1
vote
1answer
26 views
Symmetry of Plancherel measure (for $S_n$)
For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$:
$$
\begin{pmatrix}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 ...
1
vote
1answer
28 views
Probablistic approach
I have been reading about probabilistic approach in some problems in particular when we want to prove that something exists without explicitly constructing it. I really want to see more of this. Does ...
0
votes
1answer
26 views
Integrate over the uniform distribution on the simplex
Let $p=(p_1,\ldots,p_n)$ correspond to points in a simplex that add up to one, i.e. $p$ is a discrete probability distribution. I would like to compute an integral of the form $\int dp_1\ldots\int ...
0
votes
0answers
42 views
What does commensurable in this context mean?
Suppose that we observe a finite set $T\subseteq U$ where $U$ can be both finite or infinite. Further we assume that a proability measure $P_T$ is defined on all the subsets $S\subseteq T$.
Then by ...
1
vote
2answers
66 views
$\mathcal{A}\perp_\mathcal{G}\mathcal{B}\wedge\mathcal{H}\subseteq\mathcal{G}\implies\mathcal{A}\perp_\mathcal{H}\mathcal{B}$?
If $\mathcal{A}\perp_\mathcal{G}\mathcal{B}$ and $\mathcal{H}\subseteq\mathcal{G}$, is it the case that $\mathcal{A}\perp_\mathcal{H}\mathcal{B}$? Here $\mathcal{A}$, $\mathcal{B}$, $\mathcal{G}$ and ...
2
votes
0answers
48 views
Proving that $ \int \left| f-g \right|~d\mu = 2\int_{A_0} (f-g)~d\mu$
Given a (dominant) measure $\mu$, consider two probability measures $f~d\mu$ and $g~d\mu$ over $(\Omega, \mathcal F)$, I'd like to check the following reasoning for showing that
$$ \int \left| f-g ...
1
vote
0answers
22 views
limit distribution of possion distribution
Assume Xn is possion with mean $\lambda_n$ and suppose that $\lambda_n\rightarrow\infty$ as $n\rightarrow \infty$. Then how to show Xn is AN$(\lambda_n,\lambda_n)$. I've tried to use characteristic ...
2
votes
0answers
22 views
Given the variance of a zero mean random variable X, what is the largest and smallest possible value for E[exp(-jX)]
I have a question that I am really eager to know the answer. It is as follows:
Given the variance of a zero mean random variable $\mathbf{X}$ is $\sigma^2$, what is the largest and smallest possible ...
2
votes
1answer
41 views
Fun with Powerball history. What probability theories apply?
I wanted to play around with probability theories and see if I find any statistical inferences from the Powerball lottery history going back to 11/1/1997. What theories could I apply besides simple ...
1
vote
0answers
26 views
Another homework question in ergodic theory
Again the source is http://www.math.ucla.edu/~biskup/275c.1.13s/PDFs/HW1.pdf this time I'm looking at #6 the part that is left as an open-ended question. If $f \in L^1$ and $\phi$ is a measure ...
1
vote
0answers
35 views
Homework questions in ergodic theory
Let $X_1, X_2, ...$ be iid. If $f: \mathbb{R}^\mathbb{N} \rightarrow \mathbb{R}$ is measurable wrt the product structure it's $L^1$ under the distribution measure induced by the $X_i$ then why is it ...
1
vote
0answers
38 views
Almost Surely Convergence
I need some help with computing the lim inf and lim sup of $ \frac 1n \sum_i X_n$ where the density of variable $X_n$ is absolute continuous, say, f(x) = exp(-x). I am interested in using the ...
0
votes
1answer
58 views
Computing PDF of Products of Two Random Variables
I've been stuck on this problem for some days. I'm hoping someone would help by chipping in a few comments. I have two i.i.d. r.v.:
$$
f_X(x)=\frac{\left(1-e^{-\frac{x}{\alpha }}\right)^{\tilde{r}-1} ...
0
votes
1answer
32 views
A probability problem involving probability density function
I am going through a probability book where the following has been claimed :
If $$f_{X}(x) = \int_{-\infty}^{\infty}f_{U}(x-y)f_{V}(y)dy$$
then $f_{X}=f_{U}*f_{V}$
where $X, U, V$ are continuous ...
1
vote
2answers
29 views
$P\left( n,\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)T \right)$ Disaggregating Tail of Poisson
I have a Poisson tail $P\left( x,\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)T \right)$ which is sum of two independent Poisson distribution with rate $\lambda_1$ and $\lambda_2$. I am trying to ...
3
votes
3answers
64 views
$E[X]$ finite iff $\sum\limits_{n} P(X>an)$ converges
Show that:
$$\sum\limits_{n \in N } P(X>an) < \infty\ \text{for some}\ a > 0 \Rightarrow E[X] < \infty \Rightarrow \sum\limits_{n \in N } P(X>an) < \infty\ \text{for every}\ a > ...
1
vote
0answers
30 views
Maximum likelihood in exponential family: $\partial_{\theta,\theta} \ln L = -\mathrm{Var}(T)$, $T$ sufficient for $\theta$
I'm confused on how the second derivative of the log-likelihood is computed in an exponential family.
There is a result which says that
If $T=T(X)$ is the natural sufficient statistic for ...
1
vote
1answer
35 views
The weighted distribution function for combination of two variables
For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$.
Suppose, some machine get this random ...
2
votes
0answers
60 views
Gaussian vectors formula
I'm interested in studying the minimal conditions over the gaussian random vector $(X,Y)$ which takes values in $\mathbb R^{n+1}$ (where $X=(X_1, ..., X_n)$ ) to ensure that the following formula ...
0
votes
0answers
38 views
Intregral of exponential of Shannon Entropy Function
Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of
$F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$
...
1
vote
1answer
38 views
Finiteness of absolute moments
If $r>0$ then why is it that $E(|X|^r)<\infty$ if and only if $E(|X-a|^r)<\infty$ for every $a$. In the simplified case of a probability distribution describing $X$ then it says that
$$\int ...
0
votes
2answers
34 views
What prevents the Expectation conditional on a sub sigma-algebra to be the the variable being conditioned?
I have come to a very simple issue which I have not figured out yet.
In fact there are two questions.
Could you please help me out on this?
Let $(\Omega,\mathcal{F},P)$ be a probability space, ...
0
votes
1answer
30 views
Convergence of a sequence of RVs
Question:
A sequence of RVs $X_1,X_2 ... $ converges by distribution to a RV if $\forall x \in \Bbb R, i \in \Bbb N: P(X_i=x) \to P(X=x)$.
Prove that these series uniformly converge (so you need to ...
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
1answer
24 views
Length of life of a fire detector
The length of life of a flame detector is exponentially distributed with paramater $\lambda=0.1/year$. Die number of events which activate the flame detector in an interval with length $t$ (heat, ...
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votes
0answers
23 views
How to convert additive measure v to characteristic function?
1
how to convert characteristic function to additive measure v?
2.
and convert back to characteristic function?
characteristics function are from density function, pdf by fourier transform
1
vote
1answer
56 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 ...
5
votes
2answers
103 views
The probability of a drunk person/random walk
A drunk person wonders aimlessly along a path by going forward 1 step and backward 1 step with equal probabilities of $\frac12$.
a) After 10 steps, what is the probability that he has moved 2 steps ...
6
votes
2answers
115 views
It is possible to define our intuitive notion for probability in subsets of $[0,1]$
I've always heard and read the sentence:
If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$.
What is the meaning for that? Is this the "real" ...
2
votes
1answer
75 views
Selecting uncountably many random numbers from (0,1) [closed]
If uncountably many numbers are selected uniformly at random from $(0,1)$, and put in the initially empty set S, then;
What is the probability S contains $1/\pi$?
What is the probability S contains ...
1
vote
0answers
28 views
Truchet tiles on a cube [duplicate]
We randomly place copies of the tiles into faces of the flattened cube.
1.Find the probability that the circular arcs on the Truchet tiles will form one loop, two loops, three loops and four loops?
...
1
vote
1answer
22 views
Mean number of particle present in the system: birth-death process, $E(X_t|X_0=i)$, $b_i=\frac{b}{i+1}$, $d_i=d$
Let $\{X_t\}$ be a birth–and–death process with birth rate
$$
b_i = \frac{b}{i+1},
$$
when $i$ particle are in the system, and a constant death rate
$$
d_i=d.
$$
Find the expected number of particle ...
1
vote
1answer
47 views
Trouble understanding sum and product of probability distributions
Having trouble understanding where can we use the sum and product of probability distributions. Could someone please provide me with a real-life scenario? I think this is what I need to understand the ...
0
votes
1answer
22 views
Martingales involving exponents
I'm trying to solve the following problem, and am having problems with the expectation operator:
Let $(X_n)_{n\geq1}$ be independent such that $E(X_i)=m_i$, $var(X_i)=\sigma_i^2$, $i\geq1$. Let ...
