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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6 views

Existence of Joint Distribution from Overlapping Marginal Distribution

Suppose $x_i\in \mathbb{R}^{n_i}$ for $i=0,1,...,k$. For each $i=1,...,k$, suppose $F_i$ is a probability measure of $(x_0,x_i)$ on $\mathbb{R}^{n_0 + n_i}$. Assume all $F_i$ have the same marginal ...
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2answers
31 views

Calculating the expectation of binomial distribution without calculating the summation

We know that expectation of a binomial distribution is $$\sum _{1}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){p}^{k}{\left(1-p\right)}^{n-k}k = np$$ But while proving it, it is being written ...
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1answer
21 views

Why is the Laplace tranform of the pdf of a random variable called the Laplace transform of that variable?

We know that moment generating function of a random variable $X$ is $$M(t)=E[e^{tX}]$$ and if we replace $t$ with $-s$ then we get the Laplace transform as follows: $$\int_{-\infty}^{\infty}e^{-sx}f(x)...
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1answer
16 views

Joint probability density function $(X^2,Y^2)$

Let $X$ and $Y$ be ramdom variables having the following joint probability density function $f(x,y)=\begin{cases} \frac{3}{8}xy & x\geq0,\,y\geq0,\:x+y\leq2\\ 0 & otherwise \end{cases}$ ...
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0answers
27 views
+50

Quantizer Functions

Let $Y \sim P_Y$ with variance $P^{\alpha_1}$ $P>1$. Assume $n \sim P_n$ with variance $P^{\alpha_2}$ for any $\alpha_2 \le \alpha_1$. Let $\mathcal{Y}$ be the set over which the random variables $...
0
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0answers
33 views

Probability of picking an integer among rationals

Intuitively, it should be zero. But there is a bijection between $\mathbb Z$ and $\mathbb Z^c$ (non-integer rationals), so, one may think that the probability is $\frac 12$. Of course, this is not ...
0
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2answers
26 views

Value of c so that $c(2-|x|-|y|)$ is a probability distribution function(see picture)

Hint: Use the formula of volume of pyaramid. My approach: I know that the integral of a pdf from $-\infty to +\infty$ gives you $1$. I tried taking the double integral, but got stuck in as how to ...
11
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1answer
123 views

Is there a probability measure on $[0,1]$ with no subsets with measure $\frac{1}{2}$?

I have a decidedly weird question. Does there exist a probability measure $(\mu, \mathcal{F})$ on $[0,1]$ such that 1) $\mu(x) = 0$ for every $x \in [0,1]$ 2) For every $r \in [0,1] \setminus \...
4
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1answer
35 views

A proof by René Schilling that a continuous Lévy process is integrable

In his treatise "An Introduction to Lévy and Feller Processes" (arXiv link), Prof. Dr. René Schilling gives a short and seemingly straightforward proof for the claim that a continuous Lévy process is ...
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0answers
20 views

In search for a sequence of r.v. with particular conditions

I am looking for a sequence of real random variable $(X_n)_{n\geq 0}$ on a probability space $(\Omega,\mathcal F, \mathbb P)$ such that : $X_{n+1}-X_{n} <+\infty$ a.s. It exists $\omega$ s.t. $...
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1answer
25 views

Conditional expectation of the product of two random variables

Suppose that $X$ and $Y$ are random variables defined on $(\Omega, \mathcal{F}, \mathbb{P})$, and let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}$. The tower property of conditional ...
2
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1answer
26 views

Expected winnings in betting game [closed]

Suppose you are playing a game where you are betting dollars and if you flip a coin and it is heads, then you win that amount, but if it's tails, you lose that amount. You use the strategy that you ...
1
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2answers
742 views

Expected gain or loss in roulette

The questions reads: There are $37$ numbers, from $0$ to $36$. Each number has an equal chance of turning up. Zero is green in color and odd numbers are in black and even numbers are in red. If ...
1
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1answer
53 views

Is this betting game profitable?

I'm wondering whether a specific betting game is profitable but I'm not quite sure how to analyse it, some good tips on how to start would be great. Suppose a fair coin is tossed repeatedly. ...
2
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1answer
22 views

Understanding the Skorohod-space

I am having a lack of understanding the Skorhodspace considering cadlag processes. A random variable $X$ is measurable mapping between two measure spaces say $(\Omega,\mathcal{F})\mapsto (\tilde{\...
1
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0answers
26 views

PDF/CDF of max-min type random variable

For i.i.d. random variables, we may write the CDF of $t=\max(t_1,\cdots,t_N)$ as $$F_t(t)=F_{t_i}(x)^n$$ and the CDF of $x=\min(x_1,\cdots,x_N)$ as $$F_x(x)=1-(1-F_{x_i}(x))^n$$ When we have $X=\...
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1answer
50 views

An example where $E\left[\lim_{n \to \infty}X_n\right] \neq \lim_{n \to \infty}E\left[X_n\right]$

As in the title, what would be an example where $$E\left[\lim_{n \to \infty}X_n\right] \neq \lim_{n \to \infty}E\left[X_n\right]$$? with $E$ representing expectation and $X_n$ is random variable? (for ...
0
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0answers
11 views

Equivalence of different definitions of (Laplace) Green function

Fix an open set $D \subset \mathbb{R}^d$. Usually the (Laplace) Green function is defined as the solution to the boundary value problem $$ \begin{cases} \Delta u(x) = \delta_y(x) \quad x \in D\,, \\u(...
2
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0answers
21 views

Covariance Matrix of Uniform Distribution Positive Definite

Suppose that $B$ is a Lebesgue measureable subset of $\mathbb{R}^d$. Let $U$ be the uniform distribution on $B$. Let $x \sim U$, and let $M = \mathbb{E}[xx^T]$. What conditions on $B$ guarantee that $...
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2answers
30 views

measure of a set which is a subset of infinitely many subsets of probability measure space [on hold]

Let $B,A_1,A_2,....$ be the subsets of a probability measure space. If $ B \subset \bigcup A_j$, show that $m(B) \le \sum_{j=0}^\infty m(A_j)$. I have no idea as how to approach it. I do have the ...
2
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0answers
53 views
+50

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
4
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3answers
87 views

Continuous version of a Poisson R.V.

I am wondering if there is a continuous version of a Poisson random variable, that has the following two features: 1) Has a CDF that agrees with the discrete Poisson distribution on the integers, and ...
2
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4answers
67 views

Probability that the second throw of a fair die exceeds the first

A player throws an ordinary die and records the score $A$. The player then throws the die again and again records the score, $B$. if $B>A$ then we set a score for this player. What is the ...
10
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1answer
1k views

Monotone class theorem

I have some question about the Monotone Class Theorem and its application. First I state the Theorem: Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that $f_\...
0
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1answer
45 views

Bayesian Estimation: calculating an integral

I am reading a book on Bayesian filtering and I have a question regarding calculating transition density $p(X_t|X_{t-1})$. My question is how the term $p(X_t|X_{t-1}, V_{t}=v)$ is converted to the ...
2
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1answer
29 views

Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
14
votes
1answer
427 views

What happens to a random walk when we increase the probabilities of going right?

Consider a random walk on the integers where the probability of transitioning from $n$ to $n+1$ is $p_n$ (and of course, the probability of transitioning from $n$ to $n-1$ is $1-p_n$); we assume all $...
1
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0answers
19 views

Supremum of expected value over equivalent measures

I'm not sure how one can proof the following statement: We have a probability space $(\Omega, \mathbb{F}, \mathbb{P})$ and a $\mathbb{F}$-measurable random variable $X$. Furthermore we have a set of ...
0
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1answer
20 views

Is this equality holds? $\overline{F^{*2}}(x)=\int_0^x\overline{F}(x-y)dF(y)$

$X_1,X_2$ are non-negative i.i.d random variables with CDF F(x). I have a problem proving that following identity holds. $$ \frac{\overline{F^{*2}}(x)}{\overline{F}(x)}=1+\int_0^x\frac{\overline{F}(...
0
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1answer
14 views

Find the limit of the following series of normal random variables.

Let $X_1,X_2,X_3,…$ be a sequence of i.i.d. $N(\mu,1)$ random variables. Then, find $$\lim_{n\to \infty} \frac{\sqrt{\pi}}{2n}\sum_{i=1}^{n}E(|X_i-\mu|).$$ My thoughts: I don't have any rigorous way ...
0
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1answer
31 views

For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely

Question in the title: For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely My main problem is that I don't even understand what $E (Xh(Y)|Y)$ means.....
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0answers
21 views

A result about weighted-sum of uniform random variables

Let $a_1,\ldots,a_m \in \mathbb{Z}$ and $U_1,\ldots,U_m$ be independent uniform random variables taking valules in $[0,1]^d$. Let $\mathcal{Z}$ be the support of the random variable $\sum_{i=1}^m a_i ...
0
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0answers
21 views

Predict the daily usage of Bandwidth of a Network

Context: I want to predict the daily usage of bandwidth of a network (consists a number of users) based on previous use . For example, I want to predict the amount of bandwidth during 8 pm to 9pm ...
14
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1answer
886 views

Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
3
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2answers
45 views

How do I prove that for a random variable $X$, we have $P(X \le a) \le p$?

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$, $\mu = \mathbb{E}(X) = 2$, and $\max(X) \le 10$, (or $P(X \ge 10) = 0$). How can I prove the following? $$P(...
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0answers
22 views

Example of Markov property

I am reading Durrett's Probability: Theory and Examples, and trying to understand its context about Markov Chains. It is not hard to understand the theorem and proofs but when it comes to concrete ...
0
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0answers
46 views

How can I prove that for a random variable $X$, we have $P(X \le \mu) = P(X \ge \mu)$? [on hold]

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$ and $\mu = \mathbb{E}(X) = 2$. How can I prove the following? $$P(X \ge \mu) = P(X \le \mu)$$ It is also ...
2
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1answer
17 views

When is the sum of uncorrelated (not necessarily with the same distribution) r.v.'s bounded in Probabilty?

Let $v_{i},\;i=1,\cdots,N$ be such that $E\left(v_{i}\right)=0$, $E(v_{i}^{2})=1$ and $E\left(v_{i}v_{j}\right)=0\;for\;i\neq j$. So $v_{i}$'s are mean zero with unitary variance, uncorrelated and ...
1
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1answer
16 views

Chi-Squared Distribution

Let $Z_1, Z_2, Z_3$ be independent standard Normal R.V.'s. Which of the following has a Chi-Square distribution with 1 degree of freedom. $$ \begin{align} A) & & & \frac{Z_1^2, Z_2^2}{2} ...
4
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1answer
67 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all $n\...
2
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1answer
33 views

Is it true that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$?

I was wondering if we can show that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$ in general? Here $X_1$ and $X_2$ are independent but may not follow the same distribution. Any hint is much ...
2
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0answers
33 views

What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
4
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1answer
26 views

$E(X_1|X_1+X_2=k)$ increases with $k$?

$X_1$ and $X_2$ are independent, but they may not follow the same distribution. I want to know whether $E(X_1|X_1+X_2=k)$ increases with $k$. I guess this is correct, but is there a proof or counter ...
3
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1answer
126 views

Are there any modern mathematicians whose research interest is in “Probability Theory”? [closed]

I have seen professors in universities list "stochastic calculus", "stochastic analysis", "stochastic processes", "stochastic geometry" and "applied probability" as research interests, but are there ...
2
votes
1answer
474 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
0
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0answers
25 views

distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
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0answers
9 views

Gram matrix for a random variable vector space with inner product?

I am wondering if it is possible to construct a list of binary valued random variables, $\{\bf{X}_1,\bf{X}_2,\bf{X}_3\}$ and define a Gram-like matrix like \begin{bmatrix} \langle\bf{X}_1,\bf{X}_1\...
-3
votes
0answers
14 views

Prove $\frac{\partial}{\partial\rho}P(X_1 > 0, X_2 > 0) =\frac{1}{2\pi\sqrt{1-\rho^2}}$ [on hold]

Let $X_1,X_2$ have a bivariate normal distribution with zero means, unit variances, and correlation $\rho$. Show that $$ \frac{\partial}{\partial\rho}P(X_1 > 0, X_2 > 0) =\frac{1}{2\pi\sqrt{1-\...
1
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0answers
45 views

Discrete random variable whose cdf is not a step function [on hold]

Let, $(\Omega,\mathcal{F},P)$ be a probability space and $X:\Omega \rightarrow \mathbb{R}$ be a random variable. Let $F_{X} (x)$ be the cumulative distribution function of $X$. Show that if $F_{X} (x)$...
0
votes
1answer
387 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...