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 [tag:probability-...

learn more… | top users | synonyms (1)

3
votes
0answers
65 views

(Infinite hat)-guessing problem

$2$ men are playing a game: they are wearing countably infinitely many hats on their heads. The hats are either black or white with probability $\frac 12$. They see the other's man hats but cannot see ...
1
vote
0answers
17 views

Lipschitz constant/derivative of the stationary distribution of a Markov chain under perturbations in the transition kernel

I'm interested in the following question: Given a parameter $t\in \mathbb{R}$ and a column stochastic matrix $P(t)$ (i.e., $e^T P(t)=e^T$ and $P(t)_{ij}\ge 0$), calculate the Lipschitz constant of ...
3
votes
1answer
70 views

Is statistical physics background desirable for probability theory?

I am talking about higher probability viz. Brownian Motion, Ergodic Theory, Concentration, Percolation, Random Graphs, Random Matrix, etc. Going through books, I find that somehow or the other, many ...
1
vote
0answers
11 views

What is the “time change” of an adapted finite-variation stochastic process?

Let $(\Omega, \mathcal F,\mathbb P)$ be a probability space equipped with a filtration $\{\mathcal F_t:t\in\mathbb R_+\}$ satisfying the usual conditions of completeness and right-continuity. Suppose ...
4
votes
0answers
44 views

Stopping-time sigma-algebra and the case at infinity, definition question.

Assume you have a probability space $(\Omega,\mathcal{F},P)$, and you have a filtration $\{\mathcal{F}_t\}$ and a stopping time $\tau$. Then all the books I have seen define the stopping time sigma-...
2
votes
1answer
47 views

Use Poincare Recurrence to show existence of $n$

Suppose $A\subset \mathbb N$ such that $d(A)=\lim_{n\to\infty}\dfrac{|A\cap [1,n]|}{n}>0$. Then show there exists $n\in\mathbb N$ such that $\overline{d}(A\cap (A-n))>0$ where $\overline{d}(B)=\...
1
vote
1answer
28 views

Why $P\left(Y>X\right)=\sum\limits_n P\left(X=n\right) \cdot P\left(Y\geq n+1\right)$

Joint Distribution Chapter of P exam book—Discrete case. Problem 41.7 (p exam book by M. Finan) Part of the question's solution was already posted here. Michal's answer was: \begin{align} P\left(X=n\...
1
vote
1answer
30 views

expected value of game involving uniform variable and its square

I am trying to determine the expected value of the following game: Let $u$ be drawn from a uniform distribution on $[0,1]$. We write down $u$ on one side of a piece of paper and $u^2$ on the other ...
0
votes
1answer
22 views

Total Probability Theorem / Partition

In the Total Probability Theorem we assume that the sample space is partitioned into subsets. If we consider $B$ to be the sample space and $A_1$, $A_2$ to be the partition then the theorem says: $$\...
0
votes
0answers
17 views

Derive a probability distribution to transform scalars for weighted random sampling

Let's say I have a case with only two choices: $N_a$ = 20 $N_b$ = 10 I want to find a probability distribution that I can map these two values too, such that the probability of one variable being ...
8
votes
0answers
65 views

Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
0
votes
0answers
27 views

A Question About the Gaussian Distribution

I've watched a video recently for Fields medalist Cédric Villani. He brought up the Gaussian distribution and made a fun experiment(Galton's Board) that demonstrates the distribution, then he said "He ...
1
vote
2answers
40 views

Derive the value of this probability analytically

Forgive me if this question is very basic but I genuinely tried to search around including this site and could not find anything that I could adapt to my understanding. ...
0
votes
1answer
32 views

Prove that: $\frac{\Sigma_{i=1}^{n}X_i}{\sqrt{n \log n}}\rightarrow N(0,\sigma^2)$ in distribution. [closed]

Let $X_1,X_2,X_3,...$be i.i.d with density $$f(x)=\begin{cases}|x|^{-3} \text{ if |x|>1}\\0\text{ otherwise}\end{cases}$$ Prove that: $\frac{\Sigma_{i=1}^{n}X_i}{\sqrt{n \log n}}\rightarrow N(0,\...
2
votes
2answers
29 views

$\{A \subset X: \chi_A \in \mathcal{F}\}$ is a sigma algebra

Suppose $\mathcal{F}$ is a collection of real-valued functions on $X$ such that the constant functions are in $\mathcal{F}$ and $f + g$, $fg$, and $cf$ are in $\mathcal{F}$ whenever $f$, $g \in \...
2
votes
0answers
33 views

A problem about the supremum of countable stopping times

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with a countable filtration $F=\left\{ \mathcal{F}_0,\mathcal{F}_1,\cdots,\mathcal{F}_n,\cdots \right\}$. $\left\{ T_n \...
2
votes
1answer
45 views

a.s. convergence and conditional expectation

I have a stochastic process $X(t,\omega)$ which is a martingale. It is showed that there exists a r.v. $X(\infty)$ such that in $L^1(\Omega)$, $\lim_{t \rightarrow \infty}X(t) = X(\infty)$. In my ...
1
vote
1answer
27 views

Compute expected received balls from boxes

I have 6 boxes: $A,B,A',B',C \text{ and } D$. The box $A$ has $n_1$ red balls that are numbered from $1, \cdots, n_1$. The box $B$ has $n_2$ green balls that are numbered from $1, \cdots, n_2$. Make a ...
2
votes
0answers
29 views

What is probility to miss at least one test?

The probability that a teacher will give an unannounced test during any class is $\large \frac 15$. If a student is absent twice, then probability that he misses at least on test is $a) \ \...
2
votes
0answers
37 views

Conditional expectation of independent normals

As it was proved in the answer here, for $Z_1, Z_2$, two independent and identically distributed random variables. Then we have: $$ \mathbb E[Z_1\mid Z_1+Z_2] =\frac{Z_1+Z_2}{2}. $$ However, suppose ...
0
votes
1answer
40 views

When are conditional expectations equal?

As a sort of a follow-up and a generalization from a previous question, suppose that we have two independent, identically distributed random variables $X, Y$ and a third random variable $W$. Is it ...
1
vote
0answers
31 views

Weak-* compactness of probability measures on compact, non-Hausdorff space

Let $X$ be a compact topological space. How would I prove that the space of probability measures $\Delta(X)$ is weak-* compact? Without assuming $X$ being Hausdorff I can't apply to the usual Reisz ...
0
votes
0answers
95 views

convergence in distribution in Banach spaces

We let $\Omega$ be a compact metric space and consider $C(\Omega)$ to be the space of all continuous functions on $\Omega$. The dual space of $C(\Omega)$ can be seen as the set of all signed borel ...
0
votes
0answers
28 views

Conditional Expectation with linearity

Solving $E(X)$ $$=E(X-a+a)$$ (By linearity) $$=E(X-a)+E(a)$$ $$=E(X-a)+E(a)$$ $$=E(X-a)+a$$ Does this hold for all probability distributions? The place where this seems counter-intuitive to me is ...
0
votes
2answers
28 views

Conditional probability distribution and prior

In a linear Gaussian model, when I multiply a prior distribution $p(x)$ with the conditional $p(y|x)$ (here x and y are vectors), which one do I get: The joint distribution p(z) where z is the ...
0
votes
1answer
56 views

Clarification of a Probability

Suppose I have two continuous, non-negative random variables, $X$ and $Y$ and I have that $$ P(X) = P(X|Y)\cdot P(Y). $$ Can I go on and say that $$ P(X\gt z) = P(X\gt z | Y\gt z)\cdot P(Y\gt z) = ...
1
vote
1answer
59 views

If $X_n \stackrel{p, quickly}{\to} X$, then $X_n \to X$.

Probability with Martingales: Without using hint, can I just do something like this: http://math.stackexchange.com/a/1538503/140308 ? With using hint: By continuity of probability, I think ...
1
vote
0answers
64 views

Ito's Formula applied to a weird equation…

EDIT: One thing I forgot to mention before is that this is all under the $\mathbb{Q}$ measure in case that changes anything I was just wondering if someone could explain how to solve this problem. I ...
0
votes
1answer
35 views

How is immortality defined for a digraph?

A presentation on immortality $m$ of a digraph was presented almost as a sink $i\rightarrow m \leftarrow j$ somehow related on conditional independence and markov equivalence classes. I am confused ...
3
votes
0answers
53 views

Hitting times in two-dimensional case: expectation of Brownian motion at a hitting time

Consider two Brownian motions $$X_{1t}=\mu t+\sigma_1B_{1t}$$ and $$X_{2t}=\mu t+\sigma_2B_{2t}.$$ Here $B_{1t}$ and $B_{2t}$ are uncorrelated. Let $\tau_1$ and $\tau_2$ be the stopping times: \begin{...
0
votes
1answer
39 views

Can one deduce independence if the conditional expectation is a constant?

Let $X,Y$ be random variables with: $E[X|Y]=c$ and $E[Y|X]=d$, where d,c are constants in $\mathbb{R}$. Can we deduce that $X,Y$ are independent?
2
votes
0answers
16 views

Central Limit Theorem Heuristics

Surrounding the central limit theorem there exist several heuristics which say when a normal distribution is a reasonable approximation to the mean $\frac{X_1 + \cdots + X_N}{N}$ of $N$ independent (...
6
votes
0answers
65 views

Kolmogoroff 0-1 does this proof work?

I have thought at this proof of the Kolmogorov 0-1 Law varying a little the sketch found in Probability essentials (Jean Jacod, Philip Protter). My questions are Is it a valid proof? Is it a bad ...
2
votes
0answers
25 views

Showing Uniform Integrability of Random variables

Let $X_1,...,X_n$ i.i.d random variables, square integrable, and with $E[X_1]=0$. Let $Y_n = \frac{|X_1 +...+X_n|}{\sqrt{n}}$ I am trying to show that $(Y_n)$ is uniformly integrable, i.e $\...
0
votes
0answers
31 views

What is the intuition behind Uniform Integrability?

A definition of Uniform Integrability I am currently working with is that: A sequence $X_1, X_2, \ldots$ of random variables is Uniformly Integrable if: $$ \sup_n \mathbb{E}\left(|X_n|\cdot \mathbb{...
-1
votes
1answer
84 views

A proposition about a stochastically continuous process with independent increments.

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with filtration $F=\left\{ \mathcal{F_t} \right\}_{t\geqslant0}$ , and $X=\left\{ X_t \right\}_{t\geqslant0}$ is an ...
-1
votes
0answers
33 views

Birthday Problem Variant: Probability of exact number of people sharing a birthday

I'm working with some data that includes a person's date of birth. The list includes 2500+ unique individuals and using Excel it's very easy to count the number of people who share a birthday with at ...
1
vote
1answer
39 views

Solve Kolmogorov differential equations for birth-death process with constant rates

I need to solve the Kolmogorov forward equations for a birth-death process whose birth/death rates $\lambda_k,k=0,\ldots$ and $\mu_k,k=1,\ldots $ are constant, i.e., $\lambda_k=\lambda$ and $\mu_k=\mu$...
2
votes
1answer
69 views

If $(X_n)$ are Poisson independent and $\mathbb{E}[S_n]\to+\infty$ then $\frac{S_n}{\mathbb{E}[S_n]} \rightarrow 1$ almost surely

Let $X_n$ be independent Poisson random variables with $\mathbb{E}[X_n] = \lambda_n$. Define $S_n = X_1 + \dots + X_n$. Show that if $\sum \lambda_n = +\infty$, then $\frac{S_n}{\mathbb{E}[S_n]} \...
4
votes
1answer
33 views

Proof that $\mathbb{E} X^k = 0$ for all odd $k$ implies $X$ symmetric for bounded $X$ without characteristic functions

I'm working through the exercises in Terry Tao's Topics in Random Matrix Theory, and came across: Let $X$ be a bounded real random variable. Show that $X$ is symmetric if and only if $\mathbb{E}X^...
0
votes
1answer
29 views

Reference: Gaussianity of linear functional of Gaussian process

My question is similar to this one, but I'm looking for a reference rather than derivation. I've been told, inserting my own commentary in square brackets, If you take $X$ in $C([a,b])$ [i.e., $X$...
-1
votes
0answers
24 views

Compute conditional probability

If a conditional probability table is given for $P(S_t|M,E)$. How to compute the value for $P(S_t = x | M,E)$ ? where $E$ is binary (0 or 1) and $M$ is ternary ?
0
votes
0answers
44 views

$\mathbf{E}\left[\frac{(U_1+c)^2}{\max((U_1+c)^2, U_2^2)} \right] \ge \mathbf{E}\left[\frac{U_2^2}{\max((U_1+c)^2, U_2^2)} \right]$

We consider two i.i.d. random variables $U_1$ and $U_2$ such that $\mathbf{E}[U_1] = \mathbf{E}[U_2] = 0$ and $\textrm{Var}[U_1] = \textrm{Var}[U_2] < \infty$. Prove that for any $c > 0$ the ...
-2
votes
1answer
32 views

Uniform Integrability - different characterisation - prove (ii)

Probability with Martingales: For the 'only if' part assuming the hint is true, then I guess we have $\forall \varepsilon_1 > 0, \exists K \ge 0$ s.t. $$E[|X|1_{|X| > K}] < \...
-1
votes
1answer
19 views

Uniform Integrability - different characterisation - prove hint

Probability with Martingales: For the 'only if' part how to prove the hint? i'm guessing it's something to do with $$E[X 1_F] \le E[X1_{\Omega}]$$ $$= E[X 1_{|X| > K}] + E[X 1_{|X| \le K}]...
0
votes
1answer
21 views

Uniform Integrability - sufficient condition and bounded convergence theorem with weaker hypothesis

Probability with Martingales: How does the result follow? Do we choose $K = (\frac{\varepsilon}{A})^{\frac{1}{1-p}}1_{A \ne 0}$ Why do we have that inequality?
0
votes
0answers
26 views

Transformation of Laplace distribution that preserves conditional distribution

Suppose, we have a $X\sim {\rm Lap}(0,a)$ with Laplace distribution with parameter a. That is \begin{align} f_X(x)= \frac{1}{2 a}e^{-|x|/a} \end{align} Now suppose we have two independent Laplace r....
1
vote
1answer
38 views

$\lim X_n = 0$ iff $b > 0$

Probability with Martingales: It looks like $$\lim \exp\{aS_n - bn\} = 0$$ if $b > 0$ because $$\lim aS_n - bn = -\infty \tag{*}$$ but how to prove $(*)$?
3
votes
4answers
75 views

Conditional expectation of independent variables

Claim. Let $Z_1, Z_2$ be two independent and identically distributed random variables. Then we have: $$ \mathbb E[Z_1|Z_1+Z_2] =\frac{Z_1+Z_2}{2}. $$ Proof. To see this, I have proceeded as follows. ...
1
vote
0answers
14 views

“Return probability” to origin of a variant of the random walk.

Let $\{\epsilon_t\}_{t\ge0}$ be an iid sequence of random variables and let $\lambda>1$. I am interested in the following process: Let $X_0 = 0$ and $$ X_{t+1} = \lambda(X_t+\epsilon_t). $$ This ...