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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Very easy question for solving the following probability density function

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} ...
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11 views

Probability question involving stochastic process

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 ...
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1answer
25 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
27 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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16 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 ...
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1answer
20 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. ...
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2answers
40 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 ...
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1answer
21 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 ...
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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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14 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 ...
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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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9 views

Transient random walks on $\mathbb{Z}$ with step $\pm a_n$ of probability $2^{-n}$

Construct a transient random walks on $\mathbb{Z}$ as follows: For $n>0$, the step is $\pm a_n$ (to be chosen) with probability $2^{-n}$ . What I thought was that $a_n$ has to grow exponentially. ...
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0answers
12 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) = ...
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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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5 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 ...
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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 ...
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1answer
43 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 ...
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1answer
15 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, ...
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2answers
42 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)= ...
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25 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 ...
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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 ...
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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 $$g:=|h|^2$$
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1answer
41 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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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 ...
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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
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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 = ...
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1answer
21 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) | ...
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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 ...
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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 $F_n(x)\rightarrow F(x)$ ...
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21 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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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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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 ...
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36 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 ...
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3answers
38 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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20 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 $$m_S = \frac{(\theta_1+\theta_2)^2}{\theta_1^2+\theta_2^2}$$ ...
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1answer
29 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 ...
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20 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 ...
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1answer
10 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 ...
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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 = ...
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61 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 ...
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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 ...
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3answers
59 views
+100

Proving two random variables differ with positive probability

EDIT: Despite the help of the posters below, I'm still confused. I'm rephrasing the question slightly. Can someone hep me with rephrased problem: Suppose that $X$ is a random vector and $Y$ a random ...
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1answer
13 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
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1answer
15 views

Prove existence/non-existence of a pdf given mean, std, range

Given: Mean = 100, Range = [4, 10000], std = 3000 Is it possible to prove whether a pdf exists or not that satisfies these values? If it does exist, what would be approximate shape of the ...
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1answer
33 views

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

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random ...
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1answer
51 views

E(XY) = E(X).E(Y|X) . Is this true for mean = zero.

I know that Joint Probability density function for two random functions $X$ and $Y$ $$P(XY) = P(X)\cdot P(Y|X)\tag{1}$$ But I just read in a set of lecture notes that for E(X)=E(Y)=0 $$E(XY) = ...
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1answer
27 views

Property of submartingale and supermartingle?

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$. And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$. If it is true, then why? I feel confused because the ...
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2answers
36 views

Understanding a proof on almost sure convergence

I'm having trouble with the following proof: $\color{red}{\text{This is not the end of the proof.}}$ I'm not understanding the definition of $A_k$. For instance $A_5\nsubseteq A_6$ because $A_6$ ...