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-...

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2
votes
1answer
38 views

$X_n$ Poisson independent, $\mathbb{E}[X_n] = \lambda_n$. If $\sum \lambda_n = +\infty$, then $\frac{S_n}{\mathbb{E}[S_n]} \rightarrow 1$ a.s

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]} \...
1
vote
1answer
1k views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
3
votes
2answers
52 views

Determine whether a random binary sequence was generated by human or natural process

Given a binary sequence, how can I calculate the quality of the randomness? Following the discovery that Humans cannot consciously generate random numbers sequences, I came across an interesting ...
0
votes
0answers
18 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
17 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 $\mu_k=\mu$...
1
vote
0answers
37 views

$\langle M_{S(k) \wedge n}\rangle = A_{S(k) \wedge n}$ - definition?

Probability with Martingales: What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales? It seems that $$\...
2
votes
0answers
150 views

Kolmogorov 0-1 Law Converse?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. ...
1
vote
0answers
16 views

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

Probability with Martingales: approach 1: Assuming $$\lim E[\exp\{aS_n - bn\}] = E[\lim \exp\{aS_n - bn\}]$$ I can't seem to be able to prove $$\lim E[\exp\{aS_n - bn\}] = 0$$ with just $b &...
-1
votes
0answers
22 views

Prove $\lim M_{S(k) \wedge n}$ exists a.s. if $S(k) = \infty$. Is $N_n \ge 0$?

Probability with Martingales: Why does $\lim M_{S(k) \wedge n}$ exist a.s.? Is it connected to $$\sup E[M_{S(k) \wedge n}^2] < \infty$$ ? What I tried: My approach is to use: If $\lim ...
1
vote
0answers
47 views

Prove $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...
0
votes
0answers
22 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....
4
votes
1answer
28 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^...
2
votes
1answer
106 views

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
0
votes
1answer
15 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., $X$...
6
votes
1answer
218 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
-7
votes
0answers
36 views

Urgent question about probability of getting a huge deviation to the Law of Large Numbers [on hold]

I am looking for a maths expert to answer these questions about probability, since the answers needs to be accurate and I need the exact probability formulas written down. I have a huge bet with ...
4
votes
1answer
49 views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
0
votes
0answers
21 views

Uniform Boundary for S.D.E with Lipschitz Coefficients

Edit of progress: Since the SDE is linear, I got a solution in the form of $e^\int...$$\cdot e^\int$. By Jensen's inequality I can change the order of the left factor to have the integral on the ...
0
votes
0answers
14 views

Compute conditional probability

If a conditional probability table is given for P(St|M,E). How to compute the value for P(St = x | M,E)? where E is binary (0 or 1) and M is ternary?
2
votes
4answers
32 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. ...
0
votes
2answers
79 views

Does ergodicity imply stationarity?

In general, if a random process is ergodic, does it imply that it is also stationary in any sense?
-1
votes
0answers
14 views

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

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 ...
-1
votes
1answer
12 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}]...
-2
votes
1answer
17 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}] < \...
0
votes
1answer
15 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?
-2
votes
0answers
27 views

How to prove that probability for different initial conditions to yield similar trajectory is very small?

For $\epsilon > 0$, suppose $f$ is a function describing chaotic dynamics. Then, for any two different initial conditions, $x,y$, the trajectory obtained is by repeated application of the function $...
2
votes
0answers
11 views

Symmetry of concentration bounds on mean

Question summary: If I have a two-sided bound, can I immediately get a one-sided bound with tighter constants? Question details: Let $\mathbf X = X_1,...,X_n$ be $n$ i.i.d. real-valued random ...
2
votes
3answers
21 views

Probability of increasing order permutation

Suppose I have n elements. What's the probability of a permutation such that the first half is increasing and second half can be ordered without any constraints? (A permutation can only have distinct ...
1
vote
0answers
12 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 ...
0
votes
0answers
11 views

Question about Supermartingales

I came across the following problem: In my setting I have two sequences of non-negative integrable random variables (measurable with respect to some filtration $F_n$) which are called $X_n$ and $Y_n$. ...
5
votes
1answer
739 views

expectation and sign of the Radon-Nikodym derivative

I am new here. I have some questions about the Radon-Nikodym derivative. I hope someone is willing to help me with these. The questions are stated below. Also I added my attempts to the problem to ...
1
vote
0answers
12 views

Nash Equilibrium

Player A chooses a random integer between 1 and 100, with probability pj of choosing j (for j = 1, 2, . . . , 100). Player B guesses the number that player A picked, and receives that amount in ...
-1
votes
1answer
64 views

Someone Ripped Me Off, Please Help Calculating Odds!! [on hold]

I'm protesting a state contract, and one of the grounds for protest is that someone stole material from a past proposal my company submitted, and is representing it as their own. Besides leaving our ...
0
votes
0answers
20 views

Conditional expected value not mutually indipendent sets

Let be $E,G,H$ pairwise independent events but not mutual (e.g. $\mathbb{P}(E\cap H)=\mathbb{P}(E)\mathbb{P}(H),\,\mathbb{P}(G\cap H)=\mathbb{P}(G)\mathbb{P}(H), ...but \,\mathbb{P}(E\cap G\cap H)\ne\...
0
votes
0answers
12 views

Trouble with Bayesian Hypothesis Test Equation

A passage from Wasserman's All of Statistics: The Bayesian approach to testing involves putting a prior on $H_0$ and on the paramater $\theta$ and then computing $\mathbb{P}(H_0 \mid X^n)$. ...
0
votes
1answer
81 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
0
votes
0answers
19 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
3
votes
0answers
18 views

Factory inspections on a budget

A factory inspector is testing the efficiency of $n$ machines. To pass the inspection, each machine is required to run at or above a certain standard efficiency. The inspector can measure the ...
1
vote
0answers
26 views

ergodic theorem for expectation of positive recurrent diffusion

Suppose $X_t$ is a positive recurrent diffusion on $\mathbb{R}$ with invariant probability measure $\mu$. There is an ergodic theorem (see V.53. in Rogers & Williams volume II) that states $$\lim_{...
1
vote
1answer
40 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
1
vote
0answers
40 views

Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
3
votes
1answer
36 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
1
vote
0answers
42 views

One question regarding independence of $\pi$ systems.

Let G and H be sub-sigma-algebras of F and I and J are $\pi$ systems such that $\sigma(I)=G$ and $\sigma(J)=H$ Can anyone explain the following quote? Suppose I and J are independent, for fixed ...
4
votes
5answers
531 views

Intuition behind Chebyshev's inequality

Is there any intuition behind Chebyshev's inequality or is that only pure mathematics? What strikes me is that any random variable (whatever distribution it has) applies to that. $$ \Pr(|X-\mu|\...
1
vote
1answer
19 views

Voting with 3-way ties

From Peter Winkler's 'Mathematical puzzles' Ashford,Baxter and Campbell run for election and end up in a 3-way tie. To break it, they solicit voters' second preference and there is also a 3-way tie. ...
1
vote
2answers
40 views

X random variable in $\mathbb{N}$ independence of events

If I have a random variable $X$ with values in $\mathbb{N}$, $$\mathbb{P}(X=n)=\frac{1}{n^s\zeta(s)}$$ where $s>1$ and $\zeta$ the Riemann zeta function, then how can I show that $$A_i=E_{p_i^2}=\...
0
votes
0answers
39 views

What is the almost sure limit of the normalized sum of these random variables? [closed]

What is the almost sure limit of $\displaystyle\sum_{i=1}^n \frac {X_i} n$ if $$\displaystyle\mathbb{P}(X_n=n^2)=\frac 1 {n^2}$$ and $$\displaystyle\mathbb{P}(X_n=0)=1-\frac 1 {n^2}?$$ My guess is ...
1
vote
0answers
23 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...