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
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Let $X$ and $Y$ be square integrable random variables s.t. $E[X|Y]=Y$ and $E[Y|X]=X$. Prove $P(X=Y)=1$.

Let $X$ and $Y$ be square integrable random variables s.t. $E[X|Y]=Y$ and $E[Y|X]=X$. Prove $P(X=Y)=1$. Furthermore, when the condition changes to $X$ and $Y$ are integrable, show that the conclusion ...
0
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1answer
21 views

Let $X$ be normal with zero mean and variance $\sigma^2$, let $Y$ be uniform on $(0,\pi)$. Find the density of $Z=X+a \cos(Y)$.

Let $X$ be normal with zero mean and variance $\sigma^2$, let $Y$ be uniform on $(0,\pi)$ and let $a$ be a real number. Assume $X$ and $Y$ are independent . Find the density of $Z=X+a \cos(Y)$. I ...
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2answers
43 views

Coin Flipping - Probability and Value Proposition

Rusty with probability here... The question is: Flip a coin 11 times. If you get 8 tails or less, I will pay you \$1. Otherwise, you pay me \$7. Step 1. Find the expected value of the ...
1
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0answers
14 views

Distribution determined by its cgf

It is well known, that if the domain of the mgf $M:=E[e^{uX}]$ of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments. Consider the ...
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1answer
24 views

Intuition Behind this Theorem About Brownian Motion

I am having a hard time with the intuition behind some of the representation theorems dealing with Brownian Motion. I think if someone can simply explain the intuition behind this theorem then ...
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2answers
38 views

Compare two coin tossing games

Compare the following two games: You have a fair coin. After one toss, you will get 1 dollar if you get a head, and 0 dollars if you get a tail. How much will you be willing to pay to play this game ...
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0answers
25 views

In how many ways can weak law of large number can be proved? [on hold]

Can anyone tell me in how many ways can weak law of large numbers can be proved?(Except using Chebyshev's theorem )
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1answer
30 views

Proving $E[X]=\sum_{i=1}^{n}P(X\geq i)$ for a R.V which receives non-negative values

I'm refreshing my knowledge in probability and I cam across the following: Let $X$ be a discrete R.V that takes only non-negative values, then $E[X]=\sum_{i=1}^{n}P(X\geq i)$ I have a small ...
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4answers
77 views

Interesting probability question - husband and wife committee variation

Twenty husbands and wives (ten couples) are randomly divided into two groups. What is the probability that at exactly 4 wives are in the same group as their husbands? Attempt: There are $\binom{40}{2}...
3
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1answer
48 views
+50

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{\...
2
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1answer
26 views

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

Let $X$ and $Y$ be random 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 & \mbox{otherwise}. \end{...
3
votes
1answer
67 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 ...
0
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0answers
17 views

Sampling conditional random variables

I am going to generate a sample of random variables conditioning on a linear constraint. To make it clear, suppose that I want to generate multivariate gaussian $(0,\Sigma)$ conditioning on the plane ...
0
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2answers
20 views

3 card monte carlo variation

A friend wants to play a betting game with you. There are 3 upside-down cards on the table 2 black and 1 red. Your job is to find the red card. For every dollar you bet he will give you 2 to 1 odds (i....
2
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0answers
27 views

Infinitesimal generator of Brownian motion with additional jumps

A compound Poisson process is a jump process with two parameters, the rate of the jumps $\lambda$ and the distribution of the jumps $\mu$ ($\mu$ is a probability measure on $\mathbb{R}$). The ...
2
votes
4answers
4k views

Addition of two Binomial Distribution

What is the distribution of the variable $X$ given $$X=Y+Z$$, where $Y$~Binomial($n$, $P_Y$) and $Z$~Binomial($n$, $P_Z$)? For the special case, when $P_Y = P_Z = P$, I think that X~Binomial($2n$, $...
2
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0answers
28 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 $x \sim U$, $\mathbb{E}[x] = 0$. and let $M = \mathbb{E}[xx^T]$, be the covariance ...
1
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0answers
22 views

Conditional mutual information for continuous random variables

Cover and Thomas provides definition of Conditional Mutual Information (CMI) for discrete random variables but doesn't say anything about continuous variables. Wikipedia has a section about a "more ...
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0answers
18 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 ...
0
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0answers
13 views

Proving that a complex expression of integrals is increasing in a given parameter

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Consider any natural $n\geq3$ and any real $c$ such that $c\geq0$, and $\rho\geq0$. We want to prove ...
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1answer
47 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 : $\forall n >0,$ $X_{n+1}-X_{n} <+\infty$ a.s. It ...
0
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2answers
41 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 ...
1
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1answer
28 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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0answers
33 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 $...
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0answers
44 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
29 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
133 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
votes
1answer
39 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 ...
0
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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
27 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 ...
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2answers
743 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
54 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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0answers
34 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=\...
0
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1answer
51 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
13 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(...
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2answers
32 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 ...
4
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3answers
90 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
votes
4answers
69 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
46 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
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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
32 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 ...
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0answers
23 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
votes
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
votes
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(...