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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32 views

T-Distribution, Normal Distribution, and Confidence Intervals

In my probability class we were given the following problem: Suppose you take a sample of your friends and measure their heights. You calculate the sample mean to be 5 feet tall and the sample ...
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1answer
16 views

What metric is it in the definiton of converge in probability?

What metric or norm is it in the definiton of converge in probability ; $ \forall \epsilon, \lim_{n} \mathbb{P}(\mid X_{n}-X \mid > \epsilon) \rightarrow 0 $ as $n \rightarrow \infty$ noone seems ...
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0answers
29 views

How to prove convergence for the following?

Let $X_m$ be distributed as follows: $$\mathbb P(X_m=\pm m)=\frac{1}{2m^\beta}, \mathbb P(X_m=0)=\frac{m^\beta-1}{m^\beta}.$$ Let be $S_n=\sum_{m=1}^{n}X_m$, then I. If $\beta>1$, then ...
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1answer
63 views

$\aleph_1$ almost sure events that almost never all hold

This recent question sparked my curiosity. Is there a family of events $(E_k)_{k \in I}$ such that$\def\pp{\mathbb{P}}$ $\pp(E_k) = 1$ for any $k \in I$ but $\pp( \bigcap_{k \in I} E_k ) = 0$? Clearly ...
3
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1answer
19 views

Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the result that if $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes are ...
3
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1answer
59 views

Question regarding Brownian motion

Hello I have two questions regarding the construction of Ito's integral in Øksendals book from here: http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf On page 25 he lists these 3 properties ...
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1answer
27 views

Proving an inequality involving conditional probability

Let $(X_t)_{t\ge0}$ be a stochastic process on a probability space $(\Omega,\mathcal F, \mathbb P)$ and let $\mathcal F_t=\sigma(X_s:0\le s\le t)$. Let $\Lambda\in \mathcal F_t$ with $\Lambda\subset ...
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1answer
36 views

How to use the Markov property of Brownian motion

This is a problem from Durrett's probability with examples, exercise 8.2.1. It is not homework. The exercise states: Let $T_0 = \inf\{s > 0 : B_s = 0\}$ and let $R = \inf\{t > 1 : B_t = 0\}$. ...
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0answers
22 views

What happens when we pick a Random sample?

Let $(\Omega, \mathcal{F} , \mathbb{P})$ be a probability space and $X:\Omega \to \mathbb{R}$ be a random variable. When we simulate or pick a random sample of size $n$ from $X$, are we picking ...
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1answer
16 views

Existence of stationary solution to an autoregressive process

I have the following autoregressive process $$X_t = 5X_{t-1} - 6X_{t-2} + Z_t \tag{1}$$ where $(Z_t)_t$ is a sequence of independent standard normal random variables. Can I then find a stationary ...
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0answers
21 views

Intuition - Asymptotic Maximum Likelihood Estimator

Maximum Likelihood Estimation is quite clear to me when it is performed on finite sample sizes. The intuition of an obtained Maximum Likelihood estimate for given data $x_{1},...,x_{n}$, $n \in ...
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1answer
19 views

Incrementally adding to a uniform distribution of samples

I want to simulate the generation of objects over a defined area $A$, where the objects have a uniform spatial distribution, $D$ objects per unit area within $A$. I initially thought that, for $A$ ...
0
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2answers
32 views

Average time between successive occurrences of earthquakes?

In any given year, the probability of an earthquake greater than Magnitude $6$ occurring in the Garhwal Himalayas is $0.04$. The average time between successive occurrences of such earthquakes is ____ ...
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0answers
19 views

Conditional expectation of a hitting time of a Brownian motion and Laplace transform

I am trying to solve the following problem: Suppose B is a 1-dim Brownian motion, let $\mathcal{T}_a = inf\{t: B_t = a\}, \mathcal{T}_{a,b}=min\{\mathcal{T}_a,\mathcal{T}_b\}$. For $a < 0 < b$ ...
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0answers
26 views

Itō formula for a scalar valued function of the solution of a scalar Itō SODE

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $B$ be a real-valued $\mathcal F$-Brownian motion on $(\Omega,\mathcal ...
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1answer
45 views

M. Ross problem 12 chapter 5 - Exponential distribution

I have a question regarding problem 12(b) and (c) of chapter 5 of M.Ross "Introduction to probability models". The question is as follows: If $X_1, X_2, X_3$ are independent exponential random ...
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1answer
31 views

Finding marginal pdf of a (X,Y) triangle

Let R be the triangle in the x−y plane with corners at (−1, 0),(0, 1) and (1, 0). Assume (X, Y ) is uniformly distributed over R, that is, X and Y have a joint density which is a constant c on R, and ...
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0answers
25 views

Finding a distribution whose KL Divergence from a given distribution is a constant $\alpha$

Consider P as a multinomial distribution over k variables. I would like to find a distribution Q, also a multinomial distribution over k variables such that KL Divergence between Q from P is a ...
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2answers
41 views

Finding P(X<Y) for exponential Random Variable

Assume X and Y are independent exponential random variables with parameters $\lambda_1$ and $\lambda_2$ respectively. Compute $P(X<Y)$. I have alot of trouble solving questions like these. Is ...
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0answers
43 views

About directly finding the PDF vs finding the PDF from a CDF

Lets consider a simple question : Say $X$ and $Y$ are two random variables which are sampling uniformly from $[0,1]$. We want to compute the PDF of $XY$. The typical approach seems to be to compute, ...
2
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0answers
15 views

A dominating $\sigma$-finite measure for all conditional distributions

Suppose $X$ and $Y$ are two real-valued random variables with a specified joint probability distribution $P_{X,Y}.$ I wish to determine if there is a $\sigma$-finite measure $\mu$ on the real line ...
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1answer
39 views

Is a collection of random variables always a random vector?

Let $X_1, \ldots, X_n$ be a collection of $n$ random variables with the same sample space $\Omega$, the same $\sigma$-algebra $\mathcal{F}$ but not identically distributed, i.e., $P(X_1 = \omega)$ is ...
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0answers
52 views

Find the probability mass function of a poisson distribution

A random sample $X_1, X_2, . . . , X_5$ is taken from a Poisson distribution with parameter λ for some λ > 0. Find the joint probability mass function in as simplified a form as possible for ($X_1, ...
5
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1answer
132 views

How does the cardinality of the set of all probability measure on a set $X$ change according to the cardinality of $X$?

I was wondering concerning the following problem: Take $X$ as a parameter space endowed with its Borel $\sigma$-algebra. What is the cardinality of $\Delta (X)$, understood as the set of all ...
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9 views

Diagonal argument involving relative compactness of densities

There is a claim from a paper which I do not understand: Let $D$ be a domain in $\mathbb{R}^d$. Let $(p^{\eta})_{\eta >0}$ be a family of densities for random variables on $(C[0,T], ...
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1answer
8 views

emmission probabilities in a hidden markov model with 2 states and an alphabet of 4 characters

I'm reading through a text that is describing how to use use hidden markov models to identify areas of biological sequences that correspond to specific biological features. It starts with a simple ...
1
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1answer
18 views

identity with Poisson process

I want to show (if it is true) that if $\tau$ is a stopping time and $\mathbb{E}\tau < \infty$ then $\mathbb{E}N_\tau = \mathbb{E}\tau$ for $N_t$ - Poisson process with parameter $1$. I started ...
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1answer
17 views

Can regressors be considered as random variables?

In the linear regression model $$y = \beta_1 X_1 + \cdots + \beta_p X_p + \varepsilon \, ,$$ can the regressors $\{X_i\}_{i \in \{1, \ldots, p\}}$ be considered as random variables? I know that what ...
3
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1answer
56 views

Probability Problem Book: Graduate Level

I'm looking for a problem book in the style of Zhang's Linear Algebra: Challenging Problems for Students to prepare for a probability qualifying exam. In particular, the desirable source must have: ...
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2answers
53 views

Law of total variance intuition

Intuitively, what's the difference between 2 following terms on the right hand side of the law of total variance? $$\operatorname{Var}(Y) = \Bbb E\left[\operatorname{Var}\left(Y\mid X\right)\right] + ...
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2answers
44 views

Proving $\sum_{i=0}^n \binom{n}{i} = 2^n$ by math induction

I am having some trouble using math induction to prove the following problem: $$\sum_{i=0}^n \binom{n}{i} = 2^n$$ Where n $\geq$ 0 I know the first thing with math induction is substitute the base ...
1
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1answer
27 views

Series of Bernoulli random variables has geometric distribution

I try to solve the following task: Let $p\in ]0,1[$ and $X_1,X_2,\dots,Y_1,Y_2,\dots$ be i.i.d. Bernoulli$(p)$ random variables. We define $N:=\min\{n\in\mathbb{N}:X_n\neq Y_n\}$ and set ...
4
votes
1answer
64 views

Set measurable with respect to one product measure but not with respect to another

For $p \in (0,1)$, let $\mu_p$ be the measure on $\{0,1\}$ given by $\mu_p(\{1\}) = 1 - \mu_p(\{0\}) = p$. We can extend $\mu_p$ to a product measure on the countably infinite product ...
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1answer
20 views

Elements of the range of a random variable that are transformed into the same element

Let $X$ be a random variable and $Y = g(X)$. Then, the range or support of $Y$ can be written as $R_Y = \{g(x) \mid x \in R_X\}$. My question is whether there is a name (or standard notation) for ...
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2answers
15 views

Relation between expectation and sample points

Suppose that the expectation of a random variable $X$ is $5$. Which of the following statements is true? There is a sample point at which $X$ has the value $5$. There is a sample point at which $X$ ...
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0answers
39 views

Calculate the result of combining three multivariate Gaussian distributions

A Bayesian derivation of the Kalman Filter was provided by Ho and Lee (1964); this paper is available as a free pdf here. As part of their derivation, they substituted three multivariate Gaussian ...
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0answers
11 views

Methodology for probability dist of function or random variables

Is there a methodology that allows us to derive a distribution of functions of random variables? How would someone approach this problem? What are the key ingredients? For example in many electrical ...
1
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1answer
22 views

Probability Of 4 Decks of cards

I have a question about probability and I would just like to make sure im correct. This is the question: 4 standard decks, if we randomly select 100 cards without replacement find the probability of ...
0
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2answers
41 views

Probability density function of the negative of a random variable (Exercise 4.1.2 from Grimmett and Stirzaker)

Find the density function of $Y = a X$, where $a > 0$, in terms of the density function of $X$. Show that the continuous random variables $X$ and $-X$ have the same distribution function if and ...
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0answers
17 views

The definition of completion of one measure with respect to a family

The following is taken from the book of Revuz and Yor more or less verbatim. If $(E,\mathcal E)$ is a measure space carrying probability measure $\mu$, the completion $\mathcal E^\mu$ is the ...
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0answers
20 views

Probability that $t^2 + 2\sqrt{x}t + y = 0 $

X and Y are independent X~Geom(P) Y~Exp($\lambda $) Compute the probability that $t^2 + 2\sqrt{x}t + y = 0 $ Steps I've taken so far: Found where the determinant of the quadratic is $\geq 0$. ...
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1answer
38 views

Probability Space and Random Variable [closed]

Let $\Omega = [0, \infty)$ and define $X$ on $\Omega$ by $X(w) = \frac{1}{1+w^2} $. For each of the following intervals, $I$, find the event $(x \in I) = \{w: X(w) \in I\}$. For $I = \left[{1\over ...
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2answers
25 views

How to solve this using probability theory?

In 7 story building 3 persons got on an empty elevator on the first floor. Each of them can get out at any floor independent of each others starting from the 2nd floor. What is the probability that ...
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0answers
22 views

Compose pdf with the random variable itself and conditional probability.

When I discussed with my classmates, we was wondering whether the following statement is correct. Given $(X_k)_k$ be a discrete real-valued stochastic process and assume $p_{X_k}$ be the strictly ...
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1answer
30 views

Simplify $E(\max(X_1+Y_1, X_2+Y_2))$ when $X_1, Y_1, X_2$, and $Y_2$ are exponentially distributed

The time until A arrives is exponentially distributed with rate $\lambda_1$, and the time until B arrives is exponentially distributed with rate $\lambda_2$. Once they arrive, they will spend ...
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0answers
59 views

Pos properties.

Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as: $$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$ When given $n\in\mathbb{N}$, this function gives the 'position' of ...
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1answer
22 views

Expectation of random variable and measure transform theorem

Let $(\Omega,B,\mu)$ be a probability space where $\Omega$ is $[0,1]$, $\mu$ the Lebesgue measure, $B$ the Borel $\sigma$-algebra of $[0,1]$ and $f(w)=1-w$ be a random variable. Let $\phi:\mathbb ...
0
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1answer
52 views

Ergodicity of stochastic process

If one can show that the process converges to a stationary process in probability, does it mean that the process is ergodic?
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2answers
59 views

How do you define a sample space with rigor?

I was reading First Course on Probability by Sheldon Ross and I came across a problem which went like this: "A customer visiting the suit department of a certain store will purchase a suit with ...
2
votes
1answer
31 views

Marginal independence v.s. joint independence

Suppose that $X$ is independent with $Y$ and is also independent with $Z$. No further assumption is made about the joint distribution of $Y$ and $Z$. Does it follow that $X$ is independent with ...