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 ...

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42 views

For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
0
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1answer
57 views

I am not given figures to answer this question. Whats the right approach?

Z is a random variable defined as the sum of N independent Bernoulli trials where the probability of each Bernoulli taking the value 1 is given by p. The number of Bernoulli trials N is itself a ...
2
votes
1answer
36 views

How to solve for the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$?

I would like to find the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$. My strategy is to use Ito's general formula with: $$ f(t, B_t) = f(0,0) + \int_0^t \frac{df}{dx}(s, B_s) dB_s + ...
1
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1answer
29 views

Why does the given condition imply the following random variables are not independent?

Let $Y ∼ U[0, 1]$ be uniformly distributed in the interval $[0, 1]$. Define the random variables $X_1, X_2$ as $$X_1 = sin(2πY )$$ $$X_2 = cos(2πY )$$ Why does the fact that $X_1^2 + X_2^2 = 1$ imply ...
-1
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0answers
23 views

How to calculate weighted negative log-likelihood? [closed]

I'm calculating the negative log-likelihood for a bunch of tasks. The tasks appear at a certain time point and I want to give to the newer tasks a higher weight. $$ NLL = p_{1} * p_{2} *... * p_{n} = ...
0
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0answers
6 views

Finding the norm of estimation error asymptotically

Let $\theta \in \mathbb{R}^p$ be such that it has uniform distribution on the set of standard unit vectors $\{\tau e_1,\ldots,\tau e_p\}$, for $\tau=\sqrt{(2-\varepsilon)\log p}, \varepsilon>0$. ...
1
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1answer
13 views

Give necessary and sufficient conditions so the sum of random variables converges almost surely

$\{X_k\}_{k}$ are independent random variables on the probability space $(\Omega, \mathcal F, P)$ and $X_k$ has gamma density $f_k(x)$ where $f_k(x)=\dfrac{x^{a_x-1}e^{-x}}{\Gamma(a_k)}$ where $x,a_k ...
0
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1answer
31 views

How to deal with a conditional binomial question involving coin flips?

Suppose that we have a sequence of fair coin flips. At each round, it is either Heads, which we denote by $H$, or tails, which we denote as $T$. Now, at the end of $N$ flips, which is assumed to be a ...
1
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1answer
19 views

Number of balls in a slot

Let $N$ balls are distributed among $r$ cells at random, each cell being free to receive any number of balls. Calculate the probability that a particular cell contains k balls ...
0
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1answer
32 views

Expected win in a lottery

In a lottery $1000$ tickets are sold and the cost of a ticket is if $10$dollars. The lottery offers a first prize of if $1,000\,$ dollars, two second prizes of $500\,$ ...
2
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1answer
33 views

Theoretical interpretation of simulating from a distribution

Suppose there is a random variable $X$ with marginal density $p_X$. However only the conditional densities $\{p_{X\mid\Theta}(\cdot\mid\theta):\theta \in \mathbf{T}\}$ are known directly, where ...
0
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1answer
12 views

Limes superior and random variables

I want to show the following: Let $X_1,X_2\dots$ be i.i.d. random variables. Let $\text{E}[|X|^p]=\infty$ for $p>0$. Show that $$P(\limsup\limits_{n\to\infty }\{|X_n|\geq n^{1/p}\})=1$$ What I ...
0
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0answers
18 views

measure-theoretic probability, (sets of) null events and non-zero probability

Assuming a well-defined probability space $ (\Bbb{R},\mathscr{B},\Bbb{P}_X) $, where $\mathscr{B}$ is the Borel $\sigma$-field, and for a random variable $X$ having a continuous probability density ...
2
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0answers
57 views
+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
2
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2answers
48 views

Showing that this is a martingale.(4.13 in Øksendals SDE)

This is an exercise from Øksendals stochastic differential equations, where I get stuck. It is exercise number 4.13.(I simplified the notation a bit.) I have that X is an Itô-process where: ...
0
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1answer
51 views

Infinite probability density?

I've read that for a "[..]random variable strongly "localized" around a single value", the probability density function (PDF) could be: $p(x)=\frac {1}{2\epsilon}$, with $\epsilon \to 0$, and ...
1
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1answer
23 views

Let $X\sim\mathcal N(0,A)$ , where $A\sim Exp(1)$. How do I recover the joint distribution for $Z=(X,A)$?

Unfortunately there is no recipe for computing the joint distribution, just the other way around (from the joint distr. to the marginal ones). Would appreciate any help to find an Ansatz for this ...
3
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0answers
37 views

Hitting Times for Brownian Motion - Levy Process?

Let $X$ be a Brownian motion and let $$H_a = \inf\{ s \ge 0 \mid X_s = a \} \;\ \text{and} \;\ S_a = \inf\{ s \ge 0 \mid X_s > a \}.$$ Now, I've shown that $H_a$ and $S_a$ are equal almost surely ...
0
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2answers
35 views

Find dependent event when two dice are thrown simultaneously.

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of ...
0
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0answers
28 views

Waiting Time Distribution

Let X be a random variable which denotes the amount of time spent in a state(say state 'I') before changing state. As X is a random variable it must have a Probability space/sample space and a sigma ...
2
votes
1answer
63 views

Show $P\left[A<Z \mid \mathcal{G} \right]=e^{-A}$ for $Z$ standard exponential and $A$ nonnegative $\mathcal G$-measurable

I have a question about exponential distribution and conditional probability. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\mathcal{G}$ be a sub $\sigma$ algebra of $\mathcal{F}$. Let ...
0
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1answer
30 views

square-root rule of time

I tried to test the square-root-rule of time for quantiles of a normal distribution. So i created with the statiscal programming language R two variables a<-rnorm(100,mean=2,sd=1) ...
1
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2answers
32 views

Probability Question - Moment Generating Function

$$ f(x) = \begin{cases} xe^{-x}, & \text{x ≥ 0} \\ 0, & \text{elsewhere} \end{cases}$$ Q: Find the Moment Generating Function of X. Hi, I was trying to solve this question by putting the ...
2
votes
3answers
64 views

Picking two random points on a disk

I try to solve the following: Pick two arbitrary points $M$ and $N$ independently on a disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2 \leq 1\}$ that is unformily inside. Let $P$ be the distance between those ...
4
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1answer
67 views
+100

About the random $\pm 1$ matrices

I was reading the paper "On the probability that a random $\pm 1$ matrix is singular". In the paper the author defined the following notations: $M_n$: a random $n\times n$ matrix with i.i.d entries ...
2
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3answers
54 views

What is a sample of a random variable?

I've tried to learn probability, one way or another, many times in the last 50 years, and finally settled on the Kolmogorov approach, where a random variable isn't described as something like "a roll ...
0
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0answers
10 views

Proposition in Daniel Kuhn et al paper “Primal and dual linear decision rules in stochastic and robust optimization”

Paper link:http://www.optimization-online.org/DB_FILE/2009/02/2218.pdf In this paper in page number 8 the autors make the following proposition: Let $\mathbb{P}$ be a probability measure on ...
2
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1answer
14 views

Every collection of measures on a compact space is uniformly tight

I am looking for a proper statement of the sentence in the title and its proof. First, let me give some context. I have a covariance stationary time series, $X$. The autocovariance function of $X$ is ...
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0answers
19 views

Function of two continuous random variables. find CDF [closed]

[\begin{array}{l}{\rm{Let X be a continuous random variable with uniform distribution on }}\left[ {0,1} \right].{\rm{ }}\\{\rm{Let Y be a continuous random variable with uniform distribution on ...
2
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0answers
20 views

Almost sure convergence and limes superior

I'm trying to prove the following exercises and I don't know if my attempts are correct. A sequence of real random variables $(X_n)$ almost surely converges to $X$ if and only if for every $\epsilon ...
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1answer
39 views

Let X be a continuous random variable with pdf… [closed]

a.) Let X be a continuous random variable with pdf $f_x(t) = \exp[-t-e^{-t}]$ for all t in the reals. Find $F_X(x)$ My solution is $$F_X(x)= P(X \le x) = ...
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1answer
13 views

Continuous and discrete random variables defined on the same probability space?

I am confused on the definition of continuous/discrete random variables defined on the same probability space. Consider the random variables $X,Y$ defined on the same probability space $(\Omega, ...
0
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0answers
13 views

Properties of Kernel Integral inner Product of Gaussian Process

Can anyone give any reference / suggest how to get the rigorous mathematical properties of the following : $$ Y=\int_{a}^{b} K_{X} (t) \ f(t) \ dt $$ where $$f \sim GP (\mu(\cdot), R ...
0
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0answers
10 views

Connect the MGF of the Survivor, Cumulative and Mass disttributions

Assume that $X$ has a known distribution $P_X$, with a generating function $\hat P_X$. What relationship links $\hat P_X$ with the MGF of X's CDF ($\hat C_X$) and SDF ($\hat S_X$). Would that ...
0
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2answers
49 views

Prove $P(X=k \mid X+Y=n) = \frac1{n+1}$

Let $f_X(k) = f_Y(k)= p(1-p)^k~$ for all $k = 0,1,2,\ldots$ for some $0 < p < 1$. Show that for any $n \ge 0$ $$P(X=k \mid X+Y=n) = \frac1{n+1}$$ for any $0 \le k \le n$. What is confusing ...
0
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0answers
9 views

Deriving the asymptotic properties of two-stage estimators

Suppose $Y_n = g(X_1, \cdots, X_n)$ is a statistic and that $\sqrt{n}(Y_n -\theta) \stackrel{d}{\to} N(0,V)$, where $\theta$ is a constant. Define $Z_{i,n} = f(X_i,Y_n)$, where $f$ is a continuously ...
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0answers
30 views

Dependence/Independence Problem in probability [closed]

Suppose a student is taught by $N$ teachers. The pdf of the marks that the student gets from $i$-th teacher is $p_{m_i}(x)$ and we assume that all $p_{m_i}(x)$'s are i.i.d i.e. ...
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0answers
78 views

Bernoulli Process [closed]

Customers depart from a bookstore according to a Bernoulli process with parameter p = 0.15 (per minute). Each customer buys a book with probability 2/3, independent of everything else. Find the ...
0
votes
1answer
38 views

If I randomly select 6 books, what is the probability I…

I have 30 books. 5 are labeled classics, 10 are labeled mysteries, 7 are labeled science, and the rest are sports. If I randomly select 6 books, what is the probability I a) select at least 2 ...
1
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0answers
47 views

Condicional Expectation when $\mathbb{E}[X] = \infty$.

Let $(\Omega, \textit{F}_0, \mathbb{P})$ and $\textit{F} \subset \textit{F}_0$. Suppose $X \geq 0$ and $\mathbb{E}[X] = \infty$. Then there is a unique $Y \textit{F}$-measurable with $0 \leq Y \leq ...
0
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2answers
30 views

Moments of a continuous random variable (Exercise 4.3.3 from Grimmett and Stirzaker)

Let $X$ be a non-negative random variable with density function $f$. Show that $$ E(X^r) = \int_0^\infty r x^{r-1} P(X > r)\,dx. $$ I tried using integration by parts to obtain \begin{align} ...
1
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1answer
23 views

An equilateral triangle has one vertex at the origin of $Oxy$ plane, one vertex at $(1;0)$, and one in the 1st quadrant. Find $cov(X;Y)$

An equilateral triangle has one vertex at the origin of $Oxy$ plane, one vertex at $(1;0)$, and one in the 1st quadrant. Suppose you choose one of these three vertices uniformly at random (i.e. each ...
2
votes
0answers
15 views

Estimate for Expectation of Reciprocal Bessel Process

Let $W=(W_{t})_{t\geq 0}$ be a standard $3$-dimensional Brownian motion, and let $a\neq 0\in\mathbb{R}^{3}$. Consider the $3$-dimensional inverse Bessel process defined by ...
0
votes
1answer
12 views

Does an embedded discrete-time Markov chain preserve its properties in continuous time?

Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a continuous ...
0
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0answers
42 views

Martingal-property of stochastic Integral w.r.t. Brownian Motion

To Show that $(e^{B_t^1}cos(B_t^2))_{t \in \mathbb{R_+}}$ (where: $B=(B_s^1,B_s^2)$ is a 2-dimensional Brownian Motion) is a Martingal I used Ito's Lemma and showed that this is equal to: $ 1+ ...
-1
votes
1answer
38 views

ergodicity in $\mathbb{Z}^d$

Fix $d \geq 1$ and let $E(\mathbb{Z}^d)$ denote the set of all edges of the graph $\mathbb{Z}^d$. Let us consider a measure preserving system $(\mathbb{R}^{E(\mathbb{Z}^d)}, ...
1
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1answer
35 views

Bernstein Inequality (wiki correction?!)

I am having trouble with one of the statements made on this wikipedia page, in particular the second Bernstein Inequality on: https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory) ...
0
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0answers
35 views

How to find approximate probability of obtaining population variance between $10$ and $15$? [closed]

A sample of $15$ observations is taken from a normal population. It has been calculated that the sample mean is 30 and the sample variance is $12.1$. Find the approximate probability of obtaining ...
0
votes
2answers
21 views

measure preserving system

Let $T$ be a measure-preserving transformation on a probability space $(\Omega, \mathcal{F}, P)$ and let $A \in \mathcal{F}$ such that $P(A) > 0$. (i) Show that there exists $n \geq 1 $such that ...
1
vote
1answer
47 views

Finding pdf of the distance between two points (My strategy right or wrong?)

I have $N$ points randomly distributed in between points $A$ and $B$ in an area. I want to find the pdf of the distance between point $A$ and $B$. Prior Knowledge: 1- $f_{d_{A,i}} \forall i \in ...