Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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3
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1answer
27 views

Intuition behind Strassen's theorem

Currently I am dissecting a proof of Strassen's theorem, which states the following: Suppose that $(X,d)$ is a separable metric space and that $\alpha,\beta>0$. If $\mathbb{P}$ and ...
1
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0answers
25 views

Let $X_n>0$ be iid and $P(X_n>t)\sim t^{-\alpha}$, show that $Y_n=n^{-1/\alpha}S_n$ and $1/Y_n$ are tight.

We are given that $X_n>0$ be iid with common distribtuon $X$, and $P(X>t)\sim t^{-\alpha}$, I need to show that the scale of $Y_n$ is $n^{1/\alpha}$. Or in other words show that ...
2
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0answers
21 views

Probability Another Stick Question [on hold]

A stick of length $1$ is broken at an arbitrary point and the leftmost piece, say $a$, is selected. Stick $a$ is then broken at an arbitrary point and the leftmost piece $b$ is selected. What is the ...
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0answers
12 views

Distribution of local time Brownian motion [on hold]

I am working on this the following problem related to Tanaka's formula. I understand the theory behind it but I have no clue where to begin. Let $B_t$ is Brownian motion, $B_0=0$ and $L_t$ is the ...
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2answers
8 views

Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
0
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0answers
9 views

Kth moment of the standard deviation from a normal population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the Kth moment of T about the origin, and state the condition for the existence ...
0
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0answers
20 views

Independence of Bernoulli r.v. and product

Let $X_1,X_2$ be independent random variables each assuming only the values $+1$ and $-1$ with probability $1/2$. Are $X_1,X_2,X_1X_2$ pairwise independent ? Are $X_1,X_2,X_1X_2$ an ...
0
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1answer
25 views

If $P(A_n) \ge \epsilon>0$ for large $n$, then $P(A_n i.o.) \ge \epsilon$

If $P(A_n) \ge \epsilon>0$ for large $n$, then $P(A_n i.o.) \ge \epsilon$. I tried mimicking the proof of B-C but it give the wrong inequity in a different direction.
2
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0answers
35 views

Does this strategy look correct to you (solving for probability density function with three Random Variables)

The following formula is a formula I got from a paper that deals with wireless network (if needed I can provide reference) $$\mathbb{P}[ X \geq T( Y+Z )] = \int_{-\infty}^{\infty} ...
0
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0answers
24 views

Probability question involving stochastic process [on hold]

A stochastic process $\{x_{k}\mid k=1,2,3,...\}$ of zeroes and ones is given with the property that $x_1 = 1, x_2 = 0$ and for every $k>2$ it is true that the probability of the event $x_k = 1$ is ...
0
votes
1answer
37 views

Prove that if $X$ is stochastically larger than $Y$ then $E(X)\ge E(Y)$

Prove that if $X$ is stochastically larger than $Y$ (i.e. $P(X > t) \ge P(Y > t)$ then $E(X)\ge E(Y)$.I understand how to solve the problem if $X$ and $Y$ are non-negative random ...
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1answer
28 views

Let A be the set of irrational numbers in [0,1]. Show that P(A)=1

Let A be the set of irrational numbers in [0,1]. Show that P(A)=1 , where P is Lebesgue measure. What ever we do there are infinite irrational numbers for every two rational numbers, right? and we ...
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0answers
20 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
0
votes
1answer
23 views

Finiteness of the number of big jumps of a Lévy process on a finite interval

Let $Y(t)$, $0 \leq t \leq 1$, be a Lévy process. Denote by $\{ \Delta Y_i \}$ the jumps of the process. I would like to show that the set $\{i: |\Delta Y_i| > r\}$ is finite almost surely. ...
0
votes
2answers
42 views

Show that Y=aX+b is an random variable.

Let X be an random variable on a given probability space and lrt a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
1
vote
1answer
22 views

Almost Surely convergence using Borell Cantelli

Let $X_n$ be a sequence of random variables. I want to show that if $E[X_n] \rightarrow C$ and $Var(X_n) \leq \frac{C}{n^2}$, where $C$ is some constant, then $X_n$ converge almost surely to $C$. I ...
0
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0answers
9 views

An example shows the difference between inference in Bayesian network and Junction Tree

Why inference in Junction tree is more efficient? There are directed graph BN and the corresponded undirected graph transformed by Junction tree algorithm. The literature describes that inference in ...
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0answers
15 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
1
vote
1answer
30 views

Question about probability in infinite game of chance

Imagine a game where you start with \$100 and toss a coin repeatedly. If it's heads - you lose \$1, if its tails - you double your money. Game ends when you lose all the money. Given infinite amount ...
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0answers
13 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
1
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0answers
21 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(x=0)$.Here is my solution... $\mu= \frac {p(2) 2!}{p(1)}$. then how can find the mean..thanks
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0answers
6 views

Autocorrelation of a Markov Chain?

Is there a general characterization of the autocorrelation metric of a Markov chain? There are some tangential issues as well: do $n$-state transition probabilities obtained through Chapman-Kolmogorov ...
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0answers
6 views

What is the transformation that maps a Gaussian distribution to a Beta distribution?

Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$. And Y is a random variable with Beta distribution over domain ...
0
votes
1answer
26 views

Equivalence of $\sigma$-algebras: generated by $[a,b]$ and $[-\infty,b]$

Show that the $\sigma$-algebras generated by the collection of all intervals of the form $[a,b]\subset\Bbb R$ and by the collection of all the intervals of the form $[-\infty,b]\subset\Bbb R$ are ...
3
votes
1answer
51 views

Confusions about Radon-Nikodym derivative and dominating measures

I have some difficulties to understand the Radon-Nikodym derivative and link it to the ordinary way of obtaining the probability density function, which is through the derivative of cumulative ...
2
votes
1answer
16 views

Martingale Conceptual Question

For a normal random walk where $Y_i = \pm\frac{1}{2}$ with equal probability and $X_i = \sum_{i=1}^n Y_i$, my book says the $\sigma$-algebra generated by a martingale is written as $\sigma(X_0, X_1, ...
3
votes
2answers
43 views

What is the Laplace transform of this random variable?

Define a random variable that takes only one value for example $$X=c$$ where c is a positive constant. What does the Laplace of it evaluate to i.e the following $$\mathcal{L}_X(s)= ...
0
votes
0answers
26 views

Convergence of $|f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)}$ on $x \in [0,1]$

I have the following relationship between $f_n$ and $f$ on $x \in [0,1]$ \begin{align*} |f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)} \end{align*} for all $x \in ...
2
votes
1answer
15 views

Convergence of the maxima of Cauchy random variables

Suppose that $(X_k)_{k \in \mathbb{N}}$ is a sequence of independent and identically distributed random variables such that $X_1$ has density function $f_{X_1}(x) = \frac{1}{\pi(1+x^2)}$, $x \in ...
2
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0answers
15 views

Distribution of the square of magnitude of a Nakagami random variable [on hold]

Given the random variable $$h \sim \operatorname {Nakagami} (m,1)$$ $$ f_{h}(h)= \frac{2m^m}{\Gamma(m)} h^{2m-1} \text{exp}(-m h^2)$$ What is the distribution of the following function $$g:=|h|^2$$
0
votes
1answer
44 views

A die is rolled 24 independent times. Let Y be the sum of the 24 resulting values. What is the variance of Y?

A die is rolled 24 independent times. Let Y be the sum of the 24 resulting values. What is the variance of Y? I know how to find the mean of Y, but I'm having some trouble finding the variance of X ...
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vote
1answer
18 views

Is there an interpretation of the Beta Distribution?

There are cases in probability where one distribution has an "interpretation" in terms of another distribution: X ~ Gamma(k,1/m) for positive integer k, can be interpreted as the distribution of ...
3
votes
2answers
28 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
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0answers
7 views

Understanding “Latent Variables”

I'm having troubles understanding the method of calculating/estimating a latent variable. I know that a latent variable is something unobserverd and therefore unknown that is ought to explain ...
3
votes
1answer
28 views

Conditional Expectation and Almost Sure Convergence

Say we have $Y \in L^2(\Omega,\mathcal{A},P)$ and that $E(Y|X) = X, E(Y^2|X) = X^2$. Then show that $Y=X$ a.s My approach: Define $\mathcal{C} = \{\omega : X(\omega) = Y(\omega)\}$. Then $Y = ...
0
votes
1answer
23 views

Finding $E[\text{min}(X_1, X_2) | X_1<X_2]$

Suppose that $X_1$ and $X_2$ are independent exponential random variables with parameter $\lambda_1$ and $\lambda_2$, find $E[\text{min}(X_1, X_2) | X_1<X_2]$ and $E[\text{max}(X_1, X_2) | ...
0
votes
2answers
33 views

Stopping Time Subset Proof

My probability textbook has a really crappy proof for the following result. Suppose $S$ and $T$ are stopping times, with $S(\omega) \le T(\omega)$ for all $\omega$. Prove that $\mathcal{F}_S \subset ...
1
vote
1answer
25 views

Convergence in Distribution

Suppose $X_n$ are $\mathbb{R}$-valued random variables. Then how would I be able to show that $X_n\rightarrow^D X$ where $X$ is also a $\mathbb{R}$-valued random variable iff $F_n(x)\rightarrow F(x)$ ...
3
votes
0answers
22 views

Laplace transform and Fourier transform of kernel

Suppose $p_t(x)=\frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$ is the Gaussian density. $p_t(x)$ is also the Green kernel of the heat equation in 1D: $(\partial_t-\frac12\Delta)u=0$. The Fourier transform ...
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0answers
15 views

On a problem of convergence of measure for Levy measures

I have a question that pertains to the Levy representation of infinitely divisible distributions. However, the technical item that is relevant to me right now is one that relates to weak or vague ...
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0answers
23 views

Comparing probabilities [on hold]

Two assumptions: The probability of being born is 1 in 400 trillion The probability of winning the lottery is 1 in 175 million How much greater (% wise) is the probability of winning the lottery as ...
2
votes
2answers
37 views

Existence of strictly positive probability measures

Let $X$ be a Hausdorff space (or let's even assume it is metrizable). A strictly positive measure on $X$ is the that gives positive measure to any non-empty open subset of $X$. Under which condition ...
0
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3answers
39 views

Formal proof that X and X squared random variables are dependent.

Intuitively I know that any $X$ and $Y = X^2$ random variables are not independent, but I can't come up with a formal proof. In the case I'm most interested in, $X$ is uniformly distributed on ...
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0answers
22 views

Questions regarding a Gamma distributed Random Variable ( first moment and square density)

Consider the following Gamma distributed RV $$\operatorname{Gamma }(m_S,\theta_S)$$ with the following shape and scale parameters $$m_S = \frac{(\theta_1+\theta_2)^2}{\theta_1^2+\theta_2^2}$$ ...
2
votes
1answer
30 views

Showing that $X_n$ ~ $N(0, a_n)$ converge to $0$ when $a_n \to 0$ sufficiently fast

If $X_n$ have distribution $N(0, a_n)$ with $\sum_{n=1}^\infty a_n^b < \infty$ for some $b > 0$, then $X_n$ converge almost surely to $0$. I was able to show (for a previous part of the ...
0
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0answers
21 views

If the difference of two i.i.d. random variables is normal, must the variables themselves be normal?

I previously asked a similar question about the sum of two i.i.d. random variables, thinking the two cases to be equivalent. But I can't see how to apply the proof of that case to this one. It is ...
0
votes
1answer
11 views

Calculate edge probability of a graph

I wonder what is the edge probability $p$ for which a random graph with $n = 5000$ nodes has the largest expected diameter? How can I calculate that? Is there someone who can help me? This would be ...
0
votes
1answer
16 views

Probability of edges in Graph

I have given a random graph G(n, p) with n = 5000 vertices and an edge probability of p = 0.004. I calculated the expected number of edges which is (0.004 * maximum number of possible edges) $pE = ...
2
votes
3answers
62 views

Poisson Process: Finding the sum of interarrivial time

One hundred items are simultaneously put on a life test. Suppose the lifetimes of the individual items are independent exponential random variables with mean $200$ hours. The test will end when ...
0
votes
1answer
11 views

what will be the PDF of the magnitude of this random variable x+j y?

if we have a complex random variable [x+j*y] where (j :sqrt(-1)) and x,y both have Gaussian distribution and statistically dependent , so what will be the distribution (PDF) of the magnitude of this ...