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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
7 views

Convergence of the expectation over random distribution

In many models, we have two step randomness. At first we choose a random distribution, then we generate a random variable according to this distribution. One related result that I saw lately is this: ...
0
votes
1answer
24 views

Let $q_n, n=1,2,3,\ldots$ be on stringing of rational numbers from $[0,1]$.

$a.) $Let $q_n, n=1,2,3,\ldots$ be on stringing of rational numbers from $[0,1]$ and let the be given a sequence of sets: $A_n=[q_n,1] n=1,2,3,\ldots$ Find $$\overline{\lim_{n\to \infty}} A_n.$$ Now ...
0
votes
1answer
22 views

Why do characteristic functions use $e^{ix}$ and not $e^{-ix}$? Does it matter?

I've heard the characteristic function be described as the Fourier-Stieltjes Transform of the distribution measure of a r.v., but I was curious as to why it's written as $E[e^{ix}]$ and not the ...
0
votes
1answer
10 views

Shannon entropy property proof

X and Y are two discrete random variables having $n$ possible values : $x_{i}(1\leq i \leq n)$ and $y_{j} (1\leq j \leq n)$. The probability mass function of X is given by $$ Pr(X=x_{i}) = p_{i}, ...
1
vote
2answers
21 views

A problem about probability.

The question is: "Maria flip a coin for $6$ times while Davide for $7$ times. What is the probability that Davide obtains more heads than Maria?" I solved this problem analysing $7$ cases: $1)$ ...
3
votes
1answer
25 views

an exercise about changing the measure and convergence in $L^1$

this is exercise 17.12 from probability essentials written by jacod & protter. Suppose $lim_{n→∞} X_n = X$ a.s. Let $Y = sup_n |X_n − X|$. Show $Y < ∞$ a.s. , and define a new probability ...
0
votes
1answer
11 views

Combinatorics-Summation doubt in the proof of the expectation of the Hypergeometric distribution.

The proof starts considering this equality: $(d/dx (1+x)^A)(1+x)^B = A(1+x)^{A+B-1}$ Then it keep on changing every $(1+x)^{A or B}$ for its binomial coefficient. That 's what I don't understand. If ...
1
vote
1answer
10 views

Squared Hellinger Distance subadditive for Product measures

How can I show that the squared Hellinger Distance is subadditive for Product measures? We have $\mathbb{P} = \otimes_{i=1}^n \mathbb{P_i}$ and $\mathbb{Q} = \otimes_{i=1}^n \mathbb{Q_i}$ ...
0
votes
1answer
19 views

How to use Bayes's rule with mixed distributions?

On page 81 of The Likelihood Principle by Berger and Wolpert (1988) I find the following claim (which references example 20 on page 75). We consider a certain statistical problem from a Bayesian ...
-3
votes
1answer
31 views

Time taken to give answer if probability is given.

This is a question that I am struggling with: Since the password is periodically changed, you would like to know the answer as soon as possible. So you decide to interrogate the minions in an order ...
1
vote
2answers
18 views

Probability of Independent Events individual vs in series

I understand that independent events (such as a fair coin flip) should not be viewed in succession. For example, if you flip heads 10 times in a row, the odds of flipping the next coin heads is still ...
1
vote
1answer
27 views

Show that the empty set is independent of $A$ for any $A$

I am somewhat stumped as to how to approach this. The only thing I can remotely think of is $$P(A\cap \emptyset) = P(\emptyset)$$ but nothing else comes to mind. Suggestions?
0
votes
1answer
30 views

Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :

$$P(X_1 < X_2 < X_3)= \frac{(1-p_1)(1-p_2)p_2p_3^2}{(1-p_2p_3)(1-p_1p_2p_3)}$$ To be frank I do not know where to start with this question, I would like an idea to get me going, or better yet ...
1
vote
0answers
16 views

Why covariance constraint subsumes the average power constraint?

I am studying an optimization problem in the form of \begin{equation} \begin{aligned} &\underset{p(x)}{\text{maximize}} & & W\\ & \text{subject to} & & 0 \preceq K_{X} ...
2
votes
1answer
20 views

Convergence in probability of product random variables

If $Y_n$s converge to constant $c$ in probability & $(X_n)$ is a sequense of random variables, is it true that $X_nY_n- cX_n$ converge to $0$ in probability? How can I prove this? Thanks in ...
2
votes
0answers
17 views

2D Brownian Motion — Does this argument work?

Consider a 2D Brownian Motion $(X_1(t),X_2(t))$ starting at $(x_1,x_2) \in \mathbb{R}^2$. Let $\tau_s = \inf \left\{t \geq 0 | X_1(s) - X_1(0) > s \right\}$, and let $Y_s = X_2(\tau_s)$. The ...
-5
votes
0answers
18 views

Calculate the variance. [on hold]

Let $\{\xi_k\}$ be a sequence of independent identically distributed random variables with mean zero-value and finite variance, $$\lim\limits_{n \to x} P\left(\frac{S_n}{\sqrt{n}}\gt 7\right)=0,371$$ ...
0
votes
1answer
22 views

Expected time of drawing all types of coins from a large pile

I've been working on the following question but am uncertain of how to solve it Consider an infinitely large pile of coins. Each coin has a number {1, 2, . . . , n} written on it, and these numbers ...
2
votes
0answers
30 views

An exercise on conditional expectation and some related questions.

I tried to solve an exercise involving conditional expectations, and in doing so some question's popped up in my mind. First the exercise: $|Z| \le c \textrm{ P.-a.s.} \Rightarrow |E\{ Z | ...
0
votes
1answer
26 views

Finding $E[W]$ and $E[W^2]$, where $W = \int_{t=0}^T B_s$ $ds$

I'm trying to find a)$E[W]$ and b) $E[W^2]$, where $W_t = \int_{t=0}^T B_s$ $ds$ ($B_s$ denotes a Brownian motion). In addition, I'd like to find $E[Z_sZ_t]$, where $Z_t = \int_0^t B_s^2$$ ds$ ...
1
vote
0answers
27 views

Hidden Markov Model Transition Probability

I am doing my assignment and I am asked to derive transition probability of a HMM. There are Three states. H, E and T. They initially gave me the information as follow. E is followed by an H 40% ...
0
votes
1answer
34 views

Expectation of the product of Brownian motions

I'm new to Stack Exchange. I'd like to find the expectation of the product of three Brownian motions: $E(B(t_1)B(t_2)B(t_3))$ I know from a previous post here that ...
1
vote
0answers
22 views

Why aren't these two sets of stochastic processes equal?

I'm learning about stochastic integrals now, and I don't understand the following: If $S$ and $L$ are two classes of processes where: $S=\{f(s,\omega) |f $ is progressively measurable and ...
-3
votes
3answers
32 views

Two random variables [on hold]

$X$ and $Y$ are two independent identically distributed random variable with pdfs given by $0.5\big[\exp (- \vert x-1\vert) \big]$, where $x$ and $y$ range from $-\infty$. to $+\infty$. Question ...
-1
votes
1answer
55 views

Independence of the components of a multidimensional Brownian motion [on hold]

Let $B = (B^1, \dots, B^n)$ be an $n$-dimensional ($n \in \{1, 2, \dots\}$) Brownian motion (i.e. $B = (B_t)_{t \geq 0} \in \Omega \rightarrow (\mathbb{R}^n)^{[0,\infty)}$ has continuous paths, $B_0 = ...
1
vote
1answer
36 views

Questions on Kolmogorov Zero-One Law Proof in Williams

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book: Here are my questions: Why exactly are $\mathfrak{K}_{\infty}$ and ...
1
vote
0answers
15 views

Convergence in distribution of distributions $p_n$ implies convergence in distribution of $s_n$?

Question Setup Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx ...
-1
votes
0answers
57 views

Almost independent vectors: where do they live on a manifold? [on hold]

I am new to this community, please don't be tough with me. My question is: Almost independent vectors - where do they live on a manifold? What about in a manifold with larger dimension? Follow-up ...
0
votes
0answers
27 views

Expectation and supremum

$X$ is a random variable. Is the following correct: $$\sup_{z}\mathbb{E}|f(X;z)|\le\mathbb{E}\sup_{z}|f(X;z)|$$ Thank's!
3
votes
1answer
21 views

Density Function of Random Variable Related to Brownian Motion

Above is my question. I've done the first two parts, that's no problem. I'm stuck on finding the density of the rv $R = W_1 / M$. I have got as far as $$g(x,y) = \frac{\partial^2}{\partial x ...
0
votes
1answer
47 views

Probability of getting extra heads

Suppose you are tossing a coin $10$ times. You observe that the first $5$ tosses result in all heads. Then what is the probability that you would get $3$ heads in the remaining $5$ tosses? I have no ...
2
votes
2answers
67 views

Expectation of $\mathbb{E}(X^{k+1})$

I have difficulties with an old exam problem : Let $X$ be a positive random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$. Show that $$\int_0^\infty t^k ...
1
vote
1answer
7 views

Countable infinite support of probabilistic measure

Let $E\subset\Omega$ be a countable infinite set. I want to define probability measure $p$ on $\Omega$, such that $p(x)>0 \iff x\in E$ and all $x\in E$ have the same probability. Is that possible? ...
2
votes
0answers
19 views

Consequence of random walk with positive speed on a graph

Consider a random walk $X(n)$ on a vertex-transitive graph where the random walk has positive speed, i.e., $$ \lim\limits_{n \rightarrow \infty} \frac{d(X(n), X(0))}{n}= \alpha>0$$ almost surely. ...
3
votes
0answers
30 views

Hitting time is a stopping time

Can somebody help me proving that the following hitting time is a stopping time? Let $\{X_t\}_{t\ge 0}$ be a real-valued, right-continuous process, adapted to a filtration $\mathfrak{F}$ which ...
0
votes
0answers
10 views

Markov chain convergence without total variation norm

Can this theorem be reformulated so that the total variation norm $\|\cdot\|_{TV}$ is not used?
0
votes
1answer
20 views

Solving a series in the proof of the expectation of the binomial distribution

I am studying the expectations and variances of the most common distributions. For the binomial distribution the mean is equal to $np$. Considering $p$ and $q$ independent variables and ...
-1
votes
0answers
26 views

Are random variables in a tail $\sigma$-algebra in the same probability space?

When defining a tail $\sigma$-algebra, are the random variables necessarily in the same probability space? I mean, it is meaningless to have one random variable in a different sample space? Need the ...
2
votes
0answers
21 views

Is Gaussian $(X_1, X_2)$ optimal for $h(a_1X_1+ a_2X_2+Z_1) - \mu \, h( b_1X_1+b_2X_2+ Z_2)$?

Let \begin{align} W &= h(X_1+Z_1) - \mu \, h( X_2+ Z_2) \quad (1) \end{align} where $h(\cdot)$ is the differential entropy function, $\mu\ge 1 $ is a scalar, and $Z_1$ and $Z_2$ are ...
5
votes
1answer
52 views

martigale convergence theorems

Let $S_n = X_{1}+\cdots + X_{n}$ be a martingale satisfying $E[X_{k}^{2}]\leq k<\infty$, for all $k$. Show that $S_{n}$ obeys the weak law of large numbers: ...
2
votes
1answer
60 views

Understanding a proof of the strong Markov property of Lévy processes

I don't understand the the last sentence of a proof of the Markov property for Lévy processes given in Jochen Wengenroth's textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008). I will appreciate ...
2
votes
1answer
17 views

Conditional probability with max(X, Y)

Let $Y_n=$ the outcome of the $n$-th die roll, let $X_{n+1} = \max \{X_n, Y_{n+1}\}$ with $X_1=Y_1$. What is $P(X_{n+1}=j \ | X_1=i_1, ..., X_n=i)$? I know that it is $P(\max \{X_n, Y_{n+1} \}=j \ | ...
1
vote
1answer
64 views

Proving a statement about probability theory

Let X be arandom variable. Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$ My answer trail: $E[|X|]=\sum_X|X|P_x(X)\lt ...
1
vote
1answer
28 views

Bayes theorem - is it applicable in any case?

I'm studying the Bayes' Theorem and I have a doubt. In this wikipedia page there's an example of application for the following events: ...
3
votes
1answer
13 views

Techniques to prove FDD convergence

When examining a sequence of stochastic processes $(\textbf{X}_n)$, $n\geq1$ convergence of marginals, i.e. $\mathbf{X}_n(t)\to\mathbf{X}(t)$ (in distribution) is often not too hard to establish for ...
1
vote
1answer
41 views

some properties of $\nu$ measure

For any given function $F$ satisfying the following properties $0\le F(x)\le1,\forall x\in\mathbb R$ $F(x)\le F(y),x\le y$ $\lim_{x\to-\infty}F(x)=0,\lim_{x\to\infty}F(x)=1$ $F$ is continuous from ...
2
votes
0answers
31 views

Hoeffding’s inequality extension

In Hoeffding’s inequality we assume that the random variables $X_i$ ,$i=1,..,n$ are i.i.d. and bounded . Is there any extension to Hoeffding’s inequality for the case that $X_i$ are identically ...
1
vote
0answers
14 views

bias reduction when the bias depends on the true parameter

Let's say we estimate a parameter, $\theta$, by $\hat{\theta}$. For this estimator we have the following property that $$\hat{\theta}\to_{p}\theta+f(\theta)$$ where $\to_{p}$ denotes convergence in ...
5
votes
1answer
50 views

A number-theoretic random walk on the integers

Suppose a random walker starts at $S_0 = 2$, and walks according to the following transition probabilities: If the walker is on the $n$th prime number $p_n$, she moves to either $p_n + 1$ or ...
2
votes
0answers
25 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...