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 ...

learn more… | top users | synonyms (1)

0
votes
2answers
51 views

Express the CDF of $Y=X^2$ [on hold]

Let $X$ be a random variable with CDF $F$. Express the CDF of $Y=X^2$ in terms of $F$.
3
votes
0answers
28 views

Generalized Binomial Model independent in the limit

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ With $0\leq \theta_n+ ...
2
votes
1answer
34 views

Simplest Solution for a Round Table Q

Company of 3 Turks, 3 British and 3 French sit at a round table. What is the probability that no two countrymans sitting next to each other? All the people are different, but sitting orders different ...
0
votes
1answer
13 views

Finding a measurable function with an independent uniform distribution

Suppose $X,Y,U$ are random variables on some probability space such that $U$ is independent of $(X,Y)$. Prove there exists a measurable function $f: \mathbb{R} \times [0,1] \rightarrow \mathbb{R}$ ...
4
votes
1answer
53 views

Probability and Quantum mechanics

I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism. To wit, we usually say that an observable ...
1
vote
2answers
39 views

Frechet-Hausdorff theorem reference from J.L. Kelley used in proof that each probability measure is inner regular

Theorem: If $S$ is a complete, separable metric space, then each probability measure on it is inner regular. Proof: Since $S$ is separable, for each $n \in \mathbb{N}$ there exist countably many ...
7
votes
3answers
127 views

Roll a fair die until a 6 appears for the third time. What is the chance that all six values have occurred?

The question in the title is a homework question that I have been stumped on for some time. My approach thus far was to treat it as an occupancy problem. From class we derived the following formula ...
4
votes
0answers
31 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
1
vote
1answer
12 views

characteristic function upper bound and uniformly continuous.

Let $X$ be a random variable and let $\phi$ be its characteristic function. Let $A$ be a nonnegative constant and consider the following inequality $$ |\phi(t)-\phi(s)| \leq \sqrt{A|1-\phi(t-s)|}. $$ ...
11
votes
2answers
256 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
3
votes
0answers
78 views
+50

Does this strategy look correct to you (solving for probability density function with three Random Variables)

The following formula is a formula I got from a paper that deals with wireless networks specifically when calculating coverage probabilities - if needed I can provide the reference- it is powerful ...
0
votes
0answers
19 views

Steps: How to derive Probability density function for geometric functions

I am not from Mathematics background and hence lack awareness of many basic knowledge. So, please pardon if this sounds too trivial. I would like to know the steps with which I can obtain the pdf of ...
2
votes
1answer
286 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
3
votes
1answer
15 views

Series of independent gaussian variables and brownian motion

I am checking the proof of the construction of a brownian motion in $[0,\pi]$. We show that \begin{gather*} t \mapsto B^m_t = \frac{t}{\sqrt{\pi}}X_0 + \sqrt{\frac{2}{\pi}}\sum_{n=1}^{2^m-1}X_n ...
1
vote
0answers
18 views

Inverse Markov inequality with additional information?

Let $X$ be a random variable with values in some finite subset of $\mathbb{N}$ containing 0. Let $a>0$ and $0<\epsilon<1$ be such that $\mathbb{P}(|X-\mathbb{E}[X]|\geq a)\leq2\epsilon\leq ...
3
votes
2answers
52 views

Existence of strictly positive probability measures

Let $X$ be a Hausdorff space (or let's even assume it is metrizable). A strictly positive measure on $X$ is the that gives positive measure to any non-empty open subset of $X$. Under which condition ...
3
votes
0answers
38 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; ...
1
vote
1answer
14 views

Combined Distribution of Random variable

How to compute $P[T1 \le T2 \le t]$ for T1, T2 is independent random variable with exponential distribution in terms of cmf, pdf of T1 and T2? Similarly for $P[T1 \le T2 \le T3.. \le t]$ ? I tried ...
1
vote
1answer
276 views

How to prove that the binomial distribution is approximately close to the normal distribution when $np(1-p) \geq 10$

I would like a formal proof for this "rule of thumb." Can you assist me in getting to this solution? I require the insights and creativities of mathematicians. We know that if $np(1-p) \geq 10$ the ...
3
votes
1answer
63 views
+100

Covariance of $Z'Vb$ given that the rows of V are i.i.d.

Suppose that we have the following entities $$ \underbrace{Z}_{n\times k},\quad\underbrace{V}_{n\times L},\quad \underbrace{b}_{L\times 1}. $$ $Z$ and $b$ are nonstochastic whereas we assume that the ...
0
votes
1answer
39 views

Equivalence of $\sigma$-algebras: generated by $[a,b]$ and $(-\infty,b]$

Show that the $\sigma$-algebras generated by the collection of all intervals of the form $[a,b]\subset\Bbb R$ and by the collection of all the intervals of the form $(-\infty,b]\subset\Bbb R$ are ...
0
votes
1answer
29 views

Martingale $X_n \to \infty$ a.s.

Construct a martingale $X_n$ such that $X_n \to \infty $ a.s. I have trouble coming up with such an example and prove it. Can someone provide an example?
0
votes
1answer
32 views

Random variable related to binomial

The number of successes $A$ in $n$ independent trials with the probability of a success is $p$ for each trial is binomially-distributed. I am interested in a scenario that adds dependence to the ...
-2
votes
1answer
32 views

How central limit theorem invoke two random variables jointly Gaussian or independent? [on hold]

$X$=$\sum_{i=1}^n d_icos\theta_i$ and $Y$=$\sum_{i=1}^n d_isin\theta_i$ If n is large enough, and $d_i$ and $\theta_i$ are independent, then X and Y are jointly Gaussian? Is it ture? and if is please ...
1
vote
1answer
16 views

Tightness of random variales

If $\{X_n\}$ is a tight family of positive r.v.s. can we say something about $\{f(X_n)\}$ where $f$ is a continuous function?
-1
votes
1answer
39 views

Let $A$ be the set of irrational numbers in $[0,1]$. Show that $P(A)=1$

Let $A$ be the set of irrational numbers in $[0,1]$. Show that $P(A)=1$ , where $P$ is Lebesgue measure. What ever we do there are infinite irrational numbers for every two rational numbers, right? ...
0
votes
0answers
21 views

How to develop a probability distribution/density function of an issue?

Assume that a health insurance company has $1000$ customers. It is estimated that the probability of a customer making a claim is $p = 0.2$ per year, independently of previous claims and other ...
-1
votes
2answers
73 views

What is the distribution of Z=min(X,Y) [on hold]

Let X and Y be independent geometric random variables. What is the distribution of Z=min(X,Y)?
0
votes
0answers
18 views

Mean time for the renewal process

The system is as below. Energy keeps coming at a node with a constant rate ρ. Node has files of size exponential(λ) to be transmitted (with fixed rate of transmission $r$, time for transmission would ...
1
vote
1answer
36 views

Sum of two random variable X and -X

Let $X$ and $Y$ be two random variables such that $X$ and $-Y$ have the same distribution. Prove that $P(X+Y<-t)=P(X+Y>t)$. I know how to prove this when $X$ and $Y$ are independent but how do I ...
0
votes
1answer
13 views

Show that the σ-algebras generated by the collection of all intervals are equivalent

Show that the σ-algebras generated by the collection of all intervals of the form [a,b]⊂R and by the collection of all the intervals of the form (−∞,b]⊂R are equivalent. i am having trouble with ...
1
vote
0answers
18 views

Adapted but not progressively measurable?

Let $X(t,\omega)$ be a stochastic process: $$ X: \mathbb{R}^+ \times \Omega \rightarrow \mathbb{R}, $$ where $(\Omega, \mathcal{F}_t, \mathcal{F}, \mu)$ is a stochastic basis. Some definitions: ...
4
votes
1answer
109 views

Find a symmetric random walk on $\mathbb{Z}$ that is transient.

I wanted to know if it is possible find a symmetric random walk on $Z$ that is not recurrent. Let $X$ have the following distribution, with a probability $1/2^{i+1}$, $X=\pm b_i$. Let ...
0
votes
1answer
16 views

A basic question on the existence of expectation?

$E\big[\sqrt(X)\big] <\infty \implies \sqrt(X) <\infty$ a.s $\implies X< \infty$ a.s $ \implies E[X] <\infty$ The expectation is computed wrt to the probability measure . So why the the ...
0
votes
2answers
23 views

Find the probability density function of $Y=X^2$

Consider the random variable X with probability density function $$f(x)=3x^2$$ if $0<x<1$, and $$f(x)=0$$ otherwise. Find the probability density function of $Y=X^2$. This is the first question ...
0
votes
2answers
18 views

Covariance of uniform distribution and it's square

I have $X$ ~ $U(-1,1)$ and $Y = X^2$ random variables, I need to calculate their covariance. My calculations are: $$ Cov(X,Y) = Cov(X,X^2) = E((X-E(X))(X^2-E(X^2))) = E(X X^2) = E(X^3) = 0 $$ because ...
0
votes
0answers
9 views

Reference requests for an opt-cited result in Jennrich (1969)

Lemma 2 on page 637 of Jennrich (1967) states that: Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. ...
0
votes
1answer
18 views

expected value of three uncorrelated random variables

Random variables ξ, η and ζ are pairwise uncorrelated. It means that E(ξ*ζ) = E(ξ)*E(ζ), etc. Is it true that in this case E(ξηζ) = EξEηEζ ? How it can be proven? Note: we don't know if they are ...
2
votes
0answers
66 views
+150

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
1
vote
0answers
28 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
7
votes
1answer
274 views

What is the most extreme set 4 or 5 nontransitive n-sided dice?

A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia) For some sets, the ...
1
vote
0answers
60 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
1
vote
0answers
55 views

Probability question involving stochastic process

A stochastic process $\{x_{k}\mid k=1,2,3,...\}$ of zeroes and ones is given with the property that $x_1 = 1, x_2 = 0$ and for every $k>2$ it is true that the probability of the event $x_k = 1$ is ...
1
vote
1answer
38 views

A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
-1
votes
2answers
64 views

Show that Y=aX+b is an random variable. [on hold]

Let X be an random variable on a given probability space and let a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
1
vote
2answers
68 views

Prove that if $X$ is stochastically larger than $Y$ then $E(X)\ge E(Y)$

Prove that if $X$ is stochastically larger than $Y$ (i.e. $P(X > t) \ge P(Y > t)$ then $E(X)\ge E(Y)$. I understand how to solve the problem if $X$ and $Y$ are non-negative random ...
0
votes
0answers
14 views

Kaplan Meier Derivation

Can someone please help me follow this proof of KP: http://data.princeton.edu/pop509/NonParametricSurvival.pdf My problems are the assumptions at the beginning: If a subject is censored at t its ...
2
votes
1answer
31 views

Almost sure convergence using Borel-Cantelli lemma

Let $(X_n)$ be a sequence of random variables. I want to show that if $E[X_n] \rightarrow C$ and $Var(X_n) \leq \frac{C}{n^2}$, where $C$ is some constant, then $X_n$ converge almost surely to $C$. I ...
1
vote
1answer
34 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(x=0)$.Here is my solution... $\mu= \frac {p(2) 2!}{p(1)}$. then how can find the mean..thanks
4
votes
1answer
39 views

Intuition behind Strassen's theorem

Currently I am dissecting a proof of Strassen's theorem, which states the following: Suppose that $(X,d)$ is a separable metric space and that $\alpha,\beta>0$. If $\mathbb{P}$ and ...