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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1answer
29 views

Probability of exactly $K$ events out of possible $N$

So I've stumbled upon this question in Grimmett and Stirzaker's text. I have their solutions manual, which starts off like this: The line above, where the statement is expanded into sums, is where ...
-2
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2answers
36 views

Exponential distribution of random variable [on hold]

Random variable $X$ has probability density function $g(x)=\frac{3}{7}x^2\mathbf{1}_{[1,2]}$. Is there a function $F: \mathbb{R}\to\mathbb{R}$ for which $F(X)$ has an exponential distribution with ...
0
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0answers
24 views

Let X and Y be jointly continuous random variables, and let $A$ be an arbitrary subset of $\mathbb{R^2}$. [on hold]

Let X and Y be jointly continuous random variables, and let $A$ be an arbitrary subset of $\mathbb{R^2}$. I want to calculate the probability that the random vector $(X, Y )$ lies in the set $A$. ...
-1
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1answer
27 views

Men and Women enter a supermarket according to independent poisson process (stochastic process) [on hold]

Men and Women enter a supermarket according to independent poisson processes having respective rates of two and four per minute. a) Starting at an arbitrary time, what is the probability that at ...
1
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1answer
30 views

Transitivity of a stochastic order

Let $X$, $Y$, $Z$ be three independent random variables such that $P(X \geq Y) \geq 1/2$, $P(Y \geq Z) \geq 1/2$. Is it true that $P(X \geq Z) \geq 1/2$? It seems true but I'm having a hard time with ...
-1
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2answers
57 views

Expected values of a dice game with a 30-sided die and a 20-sided die.

Two people, $A$ and $B$, have a $30$-sided and $20$-sided die, respectively. Each rolls their die, and the person with the highest roll wins. ($B$ also wins in the event of a tie.) The loser pays ...
0
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1answer
50 views

How to solve this integral in moment generating function

The moment generating function of generalised Pareto distribution eventually comes down to the following integral (here). $$ M_X(\theta) = \mathbb Ee^{X\theta} = \int_\mu^\infty e^{\theta ...
2
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3answers
413 views

Product of two infinite sequences

Let $p_i$ be reals in (0,1) such that $\sum_1^{\infty} p_i=\infty$ and $\sum_1^{\infty} (1-p_i)=\infty$. Prove that $\sum_1^{\infty} p_i(1-p_i)=\infty$. I know a probabilistic proof (follows from ...
1
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1answer
22 views

The hierarchy of Liapounov conditions

The general setup is an array $(X_{nj} : n \in \mathbb{N}, 1 \leq j \leq k_n)$ of random variables (where of course each $k_n$ is an integer of value at least $1$). Write $S_{n} := \sum_{j=1}^{k_n} ...
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0answers
40 views

Find probability of complex event.

Okay, so today I've had task in my test, which is supposed to be hard: Cafe serves donuts. Every day number of students eat in this cafe. $\frac{3}{5}$ of these students are engineers, and ...
2
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1answer
55 views

Generating the Borel $\sigma$-algebra on $C([0,1])$

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on ...
2
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1answer
33 views

Computer Component with Gamma Distribution?

I comes to a question of one old-exam as follows: if the life of one computer component (in year) has Gamma Distribution (if I translate correctly) with ...
1
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2answers
31 views

Two company and probability example?

I ran into a problem that seems strange to me. Two companies A,B produce a device that with probability $0.05$ and $0.01$ are broken. if we buy two devices produced by one company with equal ...
2
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2answers
44 views

Conditional expectation and absolute continuity

Let $(\Omega,\mathscr{F})$ be a measurable space and $P$, $Q$ be two probability measures. Assume $Q$ is absolutely continuous with respect to $P$ and $\mathrm{d}Q/\mathrm{d}P=f$. I will use $E^P$ and ...
10
votes
1answer
63 views

Given $\mathbb{E}[X|Y] = Y$ a.s. and $\mathbb{E}[Y|X] = X$ a.s. show $X = Y$ a.s.

Given $X,Y \in L^2(\Omega,\mathscr{F},\Bbb{P})$ such that $\mathbb{E}[X|Y] = Y$ a.s. $\mathbb{E}[Y|X] = X$ a.s. show that $\Bbb{P}(X = Y ) = 1.$ $Attempt: $ I can see that $\mathbb{E}[X|Y] = Y$ ...
4
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1answer
29 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let ...
2
votes
1answer
31 views

Distribution of $\| W_t \|^2_{L^2([0,T])}$

Motivation: consider the SDE $$dX_t = b(X_t) dt + \sqrt{\varepsilon} dW_t. \tag1$$ Consider the action, defined by $$S(\phi)=\int_0^T |\phi'(t)-b(\phi(t))|^2 dt$$ if $\phi \in H^1([0,T])$ and ...
0
votes
1answer
17 views

Independence of a random variable and a sub-$\sigma$-algebra

I am having trouble understanding one of the steps in the proof of the following lemma. Let $X$ be a random $d$-vector and $\mathcal{A}$ a sub-$\sigma$-algebra on the probability space ...
1
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1answer
35 views

Stronger version of Markov Chain

I have just started looking into the concept of Markov chains and I was wondering if anyone could help me with this problem. Let $X_1, X_2, ...$ be a Markov chain with the state space $S$. I need ...
0
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0answers
29 views

Stochastic dominance of Binomial and Poission

In order to investigate the size of the cluster of a given vetex in a random graph I need to use a fact about stochastic dominance that I don't know how to prove. Namely, I am looking for a proof of ...
0
votes
1answer
17 views

Outer Measure on a Probability Space is 1 iff its complement is null?

I am trying to prove the following theorem, which I feel should be true, but am not sure how to go about it. Suppose we are in a probability space $(\Omega, \mathcal{F}, P)$ and we define for any ...
0
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1answer
29 views

How do I compute $P(X=Y)$? for independent random variables with with geometric distribution.

let $X$ and $Y$ be independent random variables with geometric distribution and parameter $p\in(0,1)$ How do I compute $P(X=Y)$? Any help would be greatly appreciated.
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2answers
24 views

Show $\lim\limits_{m\to\infty}P(n\leq m)=1$ for some function $n:\Omega\to\mathbb{N}$

Suppose that $(\Omega,\mathcal{F},P)$ is a probability triplet and $n:\Omega\to\mathbb{N}$ is some measurable function (in particular, $n(\omega)$ is finite for each $\omega\in\Omega$). I'm ...
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0answers
8 views

Counting minimal cut sets in nonamenable graphs

Suppose that $G$ is a fixed infinite, bounded degree, countable, connected nonamenable graph, meaning that its Cheeger constant is positive. Let $x\in G$ be a fixed vertex. I need to show (for ...
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2answers
53 views

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$?

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$? I thought that by monotonicity, which states that if $A \subseteq B$ then $P(A) \le P(B)$, then: If $P(A) < P(A \cup ...
1
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0answers
40 views

Question about $M/GI/ \infty $ queue

Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...
0
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1answer
30 views

Not all finite exchangeable families can be infinitely extended

Give an example of a finite exchangeable family $(X_1, X_2, \ldots, X_n)$ which can not be extended to an infinite exchangeable family $X_1, X_2, \ldots$. Luckily, such a counterexample can be ...
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1answer
36 views

my math question is [on hold]

A student is to be selected to play a supporting role in a dram from the list of names age height(m) Bob 14 1.69 John 15 1.58 ...
-2
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0answers
20 views

Gamblers Ruin Special Case [on hold]

In "progress and pinch," the wager, initially some integer, is increased by 1 after a loss and decreased by 1 after a win, the stopping rule being to quit if the next bet is 0. Show that play is ...
1
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0answers
22 views

Convergence in distribution of BM started in (x,y) to BM started in (0,0)

Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set ...
0
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0answers
23 views

conditional expected value for multinomial distribution

Let $X = (X_1, \ldots, X_l)^T \sim Mult(n,p_1, \ldots, p_l)$. I've already calculated that $X_i \sim Bin(n,p_i), ~ i=1, \ldots l \\ P^{Z_i | X_i =k} \sim Mult(n-k, \frac{p_1}{1-p_i}, \ldots, ...
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0answers
18 views

Interpretation of conditional expectation as a random variable

I have a couple problems understanding the conditional expectation as a random variable. Consider the fair dice roll as a random variable $X$. Let $C$ be the event that the dice shows a one and ...
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0answers
95 views

Covariance inequality for infinitely many exchangeable random variables

Let $X_1, X_2, X_3,\ldots$ be exchangeable, square-integrable random variables. Show that $\mathbf{Cov}[X_1, X_2] \geq 0$. Solution: Assume that \begin{equation*} \mathbf{Cov}[X_1, X_2] ...
0
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1answer
85 views

Covariance inequality for $n$ exchangeable random variables

Let $n \in \mathbb{N}$, $n \geq 2$, assume that $X_1,\ldots, X_n$ are exchangeable, square integrable random variables with $\mathbf{E}\bigl[X^2_1\bigr] < \infty$. Prove that the following ...
0
votes
2answers
22 views

Density of conditional distribution

Let $X$ be a continuous random variable with density function $f(\cdot)$. Define $Y = 2X$, another continuous random variable. I would like to determine the conditional density of $f_{Y|X}(y|x)$. It ...
2
votes
1answer
28 views

Stochastic kernel as linear operator

Let $K$ be a stochastic kernel for a set $S$ equipped with a countably generated $\sigma$-Algebra $B(S)$, i.e. $K:S\times B(S)\rightarrow [0,1]$ such that $K(\cdot,A)$ is a measurable function for ...
2
votes
1answer
19 views

Natural structure over a set of measurable functions

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $U$ be the set of all measurable functions over $(\Omega, \mathcal{F}, \mathbb{P})$ - i.e. the elements of $U$ are all measurable ...
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0answers
15 views

Optimal $A\in \Sigma$ that maximizes an objective

Let $([0,1],\Sigma, \lambda)$ be a probability space. For any given $B\in \Sigma$, $K\in [0,1]$ and $f\in L^2(\lambda)$ with $f(x)\in[0,1]$ for all $x $, $$\max_{A\in \Sigma}\int_A f(x) d\lambda(x)- ...
0
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2answers
33 views

Does convexity of the distribution function imply convexity of the density function? [closed]

If a distribution function $F$ is convex, such that $$ \frac{\partial^2}{\partial a^2}F(x,a)\ge 0 $$ does this then imply that it density $f$ is also convex, such that $$ \frac{\partial^2}{\partial ...
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0answers
25 views

Problem on “Distribution” of conditional expectation w.r.t a sigma field [duplicate]

Let $Y$ be a random variable s.t. $E|Y|<\infty$. Let $E[Y|\mathcal{G}]$ and $Y$ have the same distribution. I need to prove that $E[Y|\mathcal{G}]=Y$ a.s. How does one use the fact that a random ...
0
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2answers
32 views

R.V. X and Y have joint density $f(x, y) = e^{(-x-y)}$, find $P(X+Y\leq 1)$

What I did was let $Z = X + Y$ and then did a double integral, where $x$ is integrated from $0$ to $1$ and $y$ is integrated from $0$ to $Z - X$, but that gave me $$-e^{-z} - e^{-1} +1$$ which is not ...
3
votes
0answers
33 views

Does the power spectral density vanish when the frequency is zero for a zero-mean process?

A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of ...
0
votes
1answer
44 views

The support of a the sum of random independent variables

The support of a continuous random variable is the set of the outcomes such that $f(x)>0$. If $X$ has support $[a,b]$ and $Y$ has support $[c,d]$ and they are independent, what is the support of ...
0
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1answer
30 views

linear least square estimation with random sum

Let $N$ be a geometric r.v. with mean $1/p$; let $A1,A2,… $be a sequence of i.i.d. random variables, all independent of $N$, with mean $1$ and variance $1$; let $B1,B2,… $be another sequence of i.i.d. ...
1
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1answer
18 views

Differentiability of CDF at 0

This might seem to be a very trivial question but anyway here we go: I'm currently reading the paper "On the Value of a Random Minimum Spanning Tree Problem" by Frieze (1984) and I'm stuck on the ...
1
vote
1answer
43 views

Proof of martingale representation theorem monotone class argument

Martingale representation theorem for reference: Theorem: (Martingale Representation) Let $M$ be a square integrable Brownian martingale with $M_0 = 0$.Then there exists a process $X$ which is ...
0
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1answer
24 views

expectation of matrices with random components

Let's say I have a matrix where some of the components are random variables. From Wikipedia, the expectation of the matrix is simply the expectation of the individual components. But, beyond that, ...
1
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0answers
27 views

Empirical distribution generates exchangeable $\sigma$-algebra

I have a problem understanding the following statement from Klenke, p. 234: If we write $\Xi_n(\omega) := \xi_n \bigl(X(\omega)\bigr) = \frac{1}{n} \sum^n_{i=1} \delta_{X_i(\omega)}$ for the ...
1
vote
1answer
27 views

Notation for a statistic, or function of a random variable

A statistic is a function of random variables, so it is also a random variable. Suppose we have a collection $X = (X_1, X_2, \dots, X_n)$, where $X:\Omega \to \mathcal{X}^n$. There are two common ...
2
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0answers
29 views

For any given value x, are there uncountably many (countably infinite) binary sequences (ones and zeroes) whose limiting relative frequency is x

I have the following question, and given few proofs (provided by friends, professors, and my myself) which seem to work, I suspect the answer is yes: But I am still not completely sure. The question ...