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-...

learn more… | top users | synonyms (1)

3
votes
1answer
75 views

Weak convergence with uniformly convergent functions.

Suppose $F_n$ are a sequence of distribution functions with the property that for any measurable $g$ , we neccesarily have that $$ \int_{\mathbb{R}} g \; dF_n \xrightarrow{n \rightarrow \infty} \int_{\...
0
votes
4answers
46 views

Two dice are rolled and the sum of the face values is less than six. What is the probability that at least one of the dice came up a three?

Two dice are rolled and the sum of the face values is less than six. What is the probability that at least one of the dice came up a three? This is just taking the initial question that I asked and ...
1
vote
2answers
91 views

Probability Question on Speed Dating

Suppose I have 2 groups of 20 people. 20 Male and 20 Female. After doing some speed dating, each Man writes down 2 women he fancies. And Each women writes down 2 men she fancies. Everybody has a ...
2
votes
2answers
88 views

Cholesky decomposition of a covariance matrix with swapped order of variables

Could you please let me know if there is a quick way to recompute result of a Cholesky decomposition of a covariance matrix, if the order of variables was switched to put a different variable as #1 on ...
5
votes
2answers
66 views

Average shortest distance between a circle and a random point lying in it

What is the average shortest distance between the circle $(x-a)^2+(y-b)^2=r^2$ and a random point lying in it? This question is just idle curiosity. Basically, it's the same as finding the ...
2
votes
0answers
22 views

Probability that there exists $M>0$ such that two processes $\{X_t\}$ and $\{Y_t\}$ are smaller than $M$ at the same time, for infinitely many $t$.

Suppose I have two iid sequences of random variables $\{X_t\}_{t\in\mathbb{N}}$ and $\{Y_t\}_{t\in\mathbb{N}}$, both absolutely continuous with full support. I know that with probability one, for any $...
0
votes
0answers
34 views

Is distribution of $Y = \sum_{n=1}^{\infty}0.5^{n} X_n$ Lebesgue measure? [closed]

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent binary random variables defined over probability space $(\Omega,\mathcal{A},P)$ such that $P(X_n = 0) = P(X_n = 1) = 0.5$. Define $Y = \sum_{n=...
1
vote
0answers
93 views

Probability of losing lotteries needed

A person earns $x_i$ amount of money every month where $x_i$ is an exponential random variable with parameter $\lambda_1$. The amount $x_i(1-p)y$, here $0 \leq p\leq 1$ and $y$ is exponentially ...
1
vote
0answers
26 views

Conditional Independence of Correlated Stochastic Processes

Let $\{X_1(t)\}$ and $\{X_2(t)\}$ be discrete-time stochastic processes, such that $X_1(t) = f(I_{t-1},\{Y_{1 t}(I_{t-1})\},\{X_{1 \tau}\}_{\tau \leq t-1} ),$ and $X_2(t) = g(I_{t-1},\{Y_{2 t}(I_{t-1})...
-1
votes
1answer
45 views

What does it mean for a function to be nonrandom? [closed]

Given a probability space, I am guessing that a random function is a function of a random variable. As such a random function is measurable under the probability space. What does it mean for a ...
0
votes
2answers
28 views

Random variables and independence of $\sigma$-algebras

If a random variable $X$ is independent from the $\sigma$-algebra $F_t$ for every $t$ in a collection of indexes, is it true that $X$ in independent from the $\sigma$-algebra generated by all the $F_t$...
0
votes
1answer
26 views

Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Probability with Martingales: To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.
-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
38 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 $$\...
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 \...
1
vote
1answer
36 views

Showing convergence of conditional probability

Let $\big(\Omega,\mathcal{F},\mathbb{P} \big)$ be a probability space, and $\big(E_n)_{n\in\mathbb{N}^*}$ such that $$ \mathbb{P}\big(E_n \mid Y \big) \underset{n\to +\infty}{\longrightarrow} 0\quad \...
0
votes
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 $A = \{A_n\}_{n \in \mathbb N}$ ...
0
votes
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 ...
0
votes
1answer
21 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 ...
1
vote
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 ...
5
votes
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}{\...
0
votes
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
votes
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. ...
0
votes
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 $E(f(x))$. ...
1
vote
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
votes
1answer
28 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 ...
0
votes
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(...
0
votes
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
votes
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
vote
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
votes
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:...
1
vote
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
vote
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
votes
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
votes
1answer
81 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
votes
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 ...
1
vote
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 $(X_i)$ ...
1
vote
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 = ...
0
votes
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
vote
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
39 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
votes
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_{...
1
vote
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,...
1
vote
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
vote
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 ...