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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1answer
29 views

Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
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0answers
10 views

scotts theorem and other representation theorems for order aggreeing qualitative quantitative probability measures

Scotts theorem and other theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, finite cancellation axioms ...
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1answer
20 views

Convergence in distribution: constant multiplication

If I have $$X_n/\sigma$$ converging in distribution to $\mathcal{N}(0,\sigma^2)$, does this mean I can just multiply through with $\sigma$ and obtain that $X_n$ converges to a standard normal? I ...
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3answers
87 views

Monty Hall Problem Intuition

I was thinking about the Monty Hall problem and I thought of a possible intuitive explanation: You choose a door. Monty gives you the option of sticking with your original choice or instead ...
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0answers
61 views

Difference between $d\mu(x)$ and $\mu(dx)$

In my lecture notes of probability course I found two different notations involving $d,\mu$ and $x$: is there any difference between $\mu(dx)$ and $d\mu(x)$? For example I read $\mu(dx) = \frac{1}{\...
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0answers
29 views

Family of distributions closed under mixture and scaling?

I am looking for a family of distributions that is closed under mixture and scaling (by scaling, I mean stretching the CDF along the horizontal axis). I have thought a lot about this, and have found ...
0
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1answer
51 views

Expected of squared uniform distribution

Say $U$ is a uniform distribution given by $U\sim\text{Unif}(0,1)$. How can I compute the $E(U^2)$. This is the definition: $\int_0^1 u^2 f_U(u)du$. In the lecture the guy takes $f_U(u)$ to be 1. ...
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1answer
56 views

Interchanging Expectation and Derivative

Suppose I have a random function, $f(x)(\omega)$. And that for fixed $\omega$, we have the derivative $g(x)(\omega)=\frac{d}{dx}f(x)(\omega)$. For a fixed $x$, I can find the expectation $E(f(x))$. ...
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1answer
37 views

Finding the distribution of a minimum of a set of variables

I have been working on a problem where $X_1,...,X_n \sim \mathrm{Exp}(\beta)$ are i.i.d. and $Y = \min\left\{X_1^{0},X_2^{X_1},X_3^{X_1+X_2},...,X_n^{\sum_{i=1}^{n-1} X_i}\right\}$ . I want to find ...
0
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1answer
27 views

Probability of error in a communication channel.

This channel takes as input a Random Variable $V$ and gives as output a Random Variable $X=V+N$ where $N$ stands for a $Standard$ $Normal$ R.V (expected value of $0$ and variance of $1$). In order ...
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0answers
14 views

Finding the CDF for $Y = e^X$ when $X \sim N(0,1)$

Problem: Let $X \sim N(0,1)$ and let $Y = e^X$. Find the CDF for $Y$. Attempted Solution: Let $y = e^x$ so that $x = \ln(y)$. Then $$ F(y) = P(Y \le y) = P(Y \le e^x) = P(X \le \ln(y)) = F_X(\ln(...
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0answers
26 views

Upper bound for conditional probability

I have a discrete Markov process $\{ X_n \}$ such that $X_n \in [0,N]$ for each $n \in \mathbb{N}$. Let $\bar{x} \in [0,N]$. I would like to prove that \begin{equation} \mathbb{P}(x_1 > \bar{x}|...
3
votes
1answer
73 views

Placing spheres uniformly at random over $\mathbb{R}^3$

Put spheres uniformly at random all over $\mathbb{R}^3$, with density 1 sphere / unit cube. All spheres have the same radius $r$. What is the probability function $p(r)$, that that there is an ...
0
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1answer
32 views

Convergence in probability of a sum of dependent random variables to 0 [closed]

Suppose we know that $A_n + B_n \xrightarrow{p} 0$ as $n \rightarrow \infty$, and we know that $A_n \xrightarrow{d} N(0,1)$. Can we say that $B_n \xrightarrow{d} N(0,1)$? Note that $A_n$ and $B_n$ are ...
1
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1answer
19 views

Tight sequence of rv's such that $V(X_n) \rightarrow +\infty$.

Let $(X_n)$ be a tight sequence of real valued rv's, i.e. $\displaystyle \lim_K\sup_n P\left(\left|X_n\right|>K\right)=0$, defined on a common probability space, such that $E\left(X_n^2\right)<+\...
3
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0answers
73 views

Revision of Borel-Cantelli, $(X_n)_{n \geq 1}$ sqn of $\geq 0$ identical RVs with $E(X) < \infty$ then $X_n/n \to 0$ a.s., are my arguments correct?

I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous. Statement:...
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1answer
26 views

Why is $M$ bounded in $\mathscr L^2$ iff $E[\lim A_n] < \infty$?

Probability with Martingales: About $(c)$ My understanding is that $(c)$ is equivalent to: $$\sup E[A_n] < \infty \iff E[\lim A_n] < \infty$$ Why is that so?
1
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1answer
40 views

Does a difference of variables generate the same Sigma-algebra?

When reading the textbook Probability and Measure, I found the below part, Note that, since $X_k=\Delta_1+\cdots+\Delta_k$ and $\Delta_k=X_k-X_{k-1}$, the sets $X_1,\ldots, X_n$ and $\Delta_1,\...
3
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1answer
66 views

Are there general methods that can be applied when using the Borel-Cantelli Lemma, to get a statement about a sequence of random variables?

I hope the title in itself is clear, if not allow me to give an example. In Class my Professor did the following: Given a sequence $(X_n)_{n \in \mathbb{N}}$ of non-negative i.i.d. RV $X_n \sim X$...
4
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1answer
80 views

conditional probabilities on densities

I have a seemingly basic question, but surprisingly my web search didn't give any satisfying answers. Let $F(s)$ be the distribution of some random variable $X$ of support $(a,b)$ with continuous ...
0
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1answer
46 views

Three-Dimensional Random Walk

A particle starts at an origin $O$ in three-space. Thinking of point $O$ as the center of a cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight ...
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0answers
31 views

Consistency in the definition of cross cumulants

Suppose that I have an $n\times 1$ random vector $X=(X_1,X_2,\ldots,X_n)'$. For $\xi=(\xi_1,\ldots,\xi_n)'\in\mathbb{R}^n$, we can define the familiar generating functions $$ M_X(\xi)=E\Big[\exp\Big(\...
2
votes
1answer
105 views

$N $ has Poisson distribution , $ X_i$ have Bernulli distribution and are independent, find the mean of $Y=X_1+\cdots+X_N$

We have a random variable $N$ that follows the Poisson distribution with parameter $\lambda$ and $Y=\sum_{i=1}^N X_i $, where $X_i$ follows the Bernoulli distribution with parameter $\rho$ and $(X_i)$ ...
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3answers
41 views

Probability Question for Random Variable $R = \sqrt{X^2 + Y^2}$

Problem: Let $(X, Y)$ be uniformly distributed on the unit disk $\{ (x,y) : x^2 + y^2 \le 1\}$. Let $R = \sqrt{X^2 + Y^2}$. Find the CDF and PDF of $R$. Attempted Solution: First note that $r \in R = ...
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0answers
17 views

For showing measurability of Brownian motion, how does this set equality holds?

It is stated that the the following set equality easily comes from continuity of paths of Brownian motion $B_t$, but I can't seem to make sense of it - $$\{(\omega,t)\in \Omega\times (0,\infty) : ...
1
vote
1answer
26 views

How can I express the minimization of the p90th percentile mathematically?

I would like to minimize the 90th percentile of a function with a normally distributed variable. If I wanted to minimize the expected value, I would do it something like this: $$ min_s \ z = E(f(X,s)...
1
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1answer
102 views

finding Expected Value for a system with N events all having exponential distribution

We have a system in which events happen one after each other. The time interval between each two events shown by random variable $t_i$. So, the time interval between the first and the second events is ...
2
votes
3answers
66 views

Finding the density for $\min\{X, Y\}$

Problem: Let $X$ and $Y$ be independent and suppose that each has a $\text{Uniform}(0,1)$ distribution. Let $Z = \min\{X, Y\}$. Find the density $f_Z(z)$ for $Z$. Hint: It might be easier to first ...
1
vote
1answer
36 views

Poisson Coin Flipping Problem

A problem from All of Statistics pg. 45: Let $N \sim \text{Poisson}(\lambda)$ and suppose we toss a coin $N$ times. Let $X$ and $Y$ be the number of heads and tails. Show that $X$ and $Y$ are ...
0
votes
1answer
38 views

Joint pdf of X and Y with absolute value

Question. Joint probability function of continuous probability X, Y is here : $f_{X,Y}(x,y) = k(|x|-|y|) \ \ \ \ \ \ \ \ \ \ (-1< y< x< 2)$ Then what is k? I mean how can I differentiate ...
0
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0answers
17 views

Measure extension

Given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},P)$ with convention $\mathcal{F}=\bigcup_{t\geq 0}\mathcal{F}_t$. Given a positive $(P,\mathcal{F}_t)$-Martignale $M_{...
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0answers
88 views

What is the value of the following probability? [closed]

Let $f_{WE}(t,\alpha,\lambda)=\alpha \lambda t^{\alpha-1}\exp\{-\lambda t^\alpha\},~ t>0.$ The joint PDF of $(X_1, X_2)$ is given as follows $$ f(x_1,x_2)=\left\lbrace \begin{array}{ll}‎ f_1(x_1,...
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0answers
11 views

On the conditional distribution of $B_{(s+t)/2}$ conditionally on $(B_t,B_s)$, for Brownian motion $B$

I've been reading stuff about Brownian motions and all that, and I came across the following statement: On proving that $B_{\frac{s+t}{2}}\sim N(\frac{x+y}{2},\frac{t-s}{4})$ conditionally on $B_s=x,...
1
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1answer
34 views

If expectation of absolute value of a RV is zero, does this mean random variable is zero a.s.?

I believe so, and was trying to prove it. Here's my attempt by contradiction: Let $A=\{\omega:|X(\omega)|>0\}$, and let $P(A)>0$. Then $$ E[|X|]=\int_{\Omega}|X|dP=\int_{A}|X|dP $$ But I am ...
3
votes
1answer
95 views

Itô-isometry in the extended case?

It is shown when constructing the Itô-integral that if: $E[\int_0^T X_t^2dt]< \infty$. Then we have that Itô-isomtry: $E[\int_0^T X_t^2dt]=E[(\int_o^TX_tdB_t)^2]$. In the extended Itô integral, ...
0
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0answers
36 views

please check my understanding about convergences with an example (probability theory)

Consider $\Omega = [0,1]$ with $\textbf{P}=U[0,1]$. Let $X\equiv 0$, $$X_1 := \textbf{1}_{[0,1/2)},X_2 := \textbf{1}_{[1/2,1)},X_3 := \textbf{1}_{[0,1/3)},X_4 := \textbf{1}_{[1/3,2/3)},X_5 := \textbf{...
0
votes
0answers
24 views

HMM limiting distribution

Consider a hidden markov model (HMM) with two hidden states $A$ and $B$ and emission support $1$ and $2$ fitted with initial state distribution $$\lambda = [\begin{array}{cc} .7&.3\end{array}]$$ ...
5
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2answers
60 views

Consider $P(A_i) = 1,\,\forall i \in \mathbb{N}.$ Prove that $P\left(\bigcap_{i=1}^{\infty} A_i \right) = 1.$

I have been working on some tough problems in my statistics book, and I came across a problem that I was having some difficulty with. Consider $$ P(A_i) = 1,\,\forall i \in \mathbb{N}.$$ I want to ...
0
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3answers
52 views

Probability problem! Extract one coin with probability 10%

If we have a box full of coins, with one gold coin with probability 10% to get extracted. Given this, is it possible to find out how many times in average I have to try extracting a coin until I get ...
2
votes
1answer
65 views

If $X$ and $Y$ are identically distributed then $(X,Y)$ and $(Y,X)$ are identically distributed?

If $X$ and $Y$ are identically distributed then $(X,Y)$ and $(Y,X)$ are identically distributed? I think the answer is no but I couldn't find a counterexample. And one more question. If $(X,Y)$ and ...
2
votes
0answers
50 views

Sum of Wiener, limit in probablitity

Show that the sequence is convergence in probability and set the limit of it: $$\sum\limits_{k=n}^{2n-1}\left(W_{(k+1)/n}^2-W_{k/n}^2-\frac{1}{n}\right)\left(W_{(k+1)/n}-W_{k/n}\right).$$ If there ...
1
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0answers
22 views

Example of a semimartingale with special properties

I have to find an example of a semimartingale X such that $\lim_{t \rightarrow \infty} X_t$ exists a.s. and $X$ is not a semimartingale up to infinity. I think it could be a deterministic function ...
0
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0answers
26 views

Method of Moments for Sequence of Bounded Random Variables

Suppose a sequence of random variables on $[0,1]$ has the property that for each $k$, the $k$'th moments converge to $\frac{1}{k+1}$. I want to show that the sequence of random variables converges in ...
3
votes
1answer
24 views

A Gambling Game

Suppose you are playing a gambling game with $n$ people (including yourself), where $n>1$. Every person involved randomly rolls an integer between $1$ and $y$, where $y>0$ and is also an integer....
1
vote
1answer
26 views

Do we not have that $X_1, \ldots, X_n \sim F$ iff $X_1, \ldots, X_n \sim f$?

From pg. 39 of All of Statistics: If $X_1, \ldots, X_n$ are independent and each has the same marginal distribution CDF $F$, we say that $X_1, \ldots, X_n$ are IID (independent and identically ...
0
votes
2answers
54 views

Expectation of the product of $2$ independent random variables

I would like some concrete examples of the expectation of the product of $2$ independent random variables. In other words $\mathbb{E}[fg]=\mathbb{E}[f]\cdot\mathbb{E}[g]$. I have learned about this in ...
0
votes
1answer
42 views

Kolmogorov's Truncation Lemma (iii)

Probability with Martingales: In the definition of $f$, is that really $z$ and not $\lceil |z| \rceil$, $\lfloor |z| \rfloor + 1$ or something? How exactly do we have the part in the $\...
1
vote
1answer
72 views

Kolmogorov's Truncation Lemma (ii)

Probability with Martingales: How exactly do we have the part in the $\color{red}{\text{red}}$ box? What I tried: $$E\left[ \sum_{n=1}^{\infty} 1_{|X| > n} \right]$$ $$ = E\left[ \...
0
votes
2answers
79 views

Strong Law of Large Numbers - Converse

Probability with Martingales: I want to try to show the last one $$\left[\limsup \frac{|S_n|}{n}\right] = \infty \ \text{a.s.}$$ which is equivalent to $$\forall k \in \mathbb N$$ $$\left[\...
5
votes
2answers
158 views

Confusion about event in a sample space.

I am a beginner in probability and counting. I am reading an open course by MIT. While reading the introductory chapter I am stuck in one conceptual doubt, if I understand correctly an event is the ...