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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Adapted but not progressively measurable?

Let $X(t,\omega)$ be a stochastic process: $$ X: \mathbb{R}^+ \times \Omega \rightarrow \mathbb{R}, $$ where $(\Omega, \mathcal{F}_t, \mathcal{F}, \mu)$ is a stochastic basis. Some definitions: ...
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Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0. $$ It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times [0,1]$. The RKHS with reproducing kernel $K$ is the ...
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characteristic function upper bound and uniformly continuous.

Let $X$ be a random variable and let $\phi$ be its characteristic function. Let $A$ be a nonnegative constant and consider the following inequality $$ |\phi(t)-\phi(s)| \leq \sqrt{A|1-\phi(t-s)|}. $$ ...
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1answer
9 views

Combined Distribution of Random variable

How to compute $P[T1 \le T2 \le t]$ for T1, T2 is independent random variable with exponential distribution in terms of cmf, pdf of T1 and T2? Similarly for $P[T1 \le T2 \le T3.. \le t]$ ? I tried ...
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2answers
12 views

Covariance of uniform distribution and it's square

I have $X$ ~ $U(-1,1)$ and $Y = X^2$ random variables, I need to calculate their covariance. My calculations are: $$ Cov(X,Y) = Cov(X,X^2) = E((X-E(X))(X^2-E(X^2))) = E(X X^2) = E(X^3) = 0 $$ because ...
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21 views

Find the probability density function of $Y=X^2$

Consider the random variable X with probability density function $$f(x)=3x^2$$ if $0<x<1$, and $$f(x)=0$$ otherwise. Find the probability density function of $Y=X^2$. This is the first question ...
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Reference requests for an opt-cited result in Jennrich (1969)

Lemma 2 on page 637 of Jennrich (1967) states that: Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. ...
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1answer
13 views

A basic question on the existence of expectation?

$E\big[\sqrt(X)\big] <\infty \implies \sqrt(X) <\infty$ a.s $\implies X< \infty$ a.s $ \implies E[X] <\infty$ The expectation is computed wrt to the probability measure . So why the the ...
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1answer
16 views

expected value of three uncorrelated random variables

Random variables ξ, η and ζ are pairwise uncorrelated. It means that E(ξ*ζ) = E(ξ)*E(ζ), etc. Is it true that in this case E(ξηζ) = EξEηEζ ? How it can be proven? Note: we don't know if they are ...
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10 views

Tightness of random variales

If $\{X_n\}$ is a tight family of positive r.v.s. can we say something about $\{f(X_n)\}$ where $f$ is a continuous function?
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18 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; ...
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9 views

Kaplan Meier Derivation

Can someone please help me follow this proof of KP: http://data.princeton.edu/pop509/NonParametricSurvival.pdf My problems are the assumptions at the beginning: If a subject is censored at t its ...
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1answer
35 views

A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
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6 views

Mean time for the renewal process

The system is as below. Energy keeps coming at a node with a constant rate ρ. Node has files of size exponential(λ) to be transmitted. At time zero, say the energy at the node be zero. So node waits ...
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1answer
35 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 ...
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36 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 ...
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24 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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2answers
20 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 ...
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35 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ 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 $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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22 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 ...
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1answer
27 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.
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65 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 networks specifically when calculating coverage probabilities - if needed I can provide the reference- it is power ...
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49 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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65 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
31 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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28 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
24 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
54 views

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

Let X be an random variable on a given probability space and let 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
30 views

Almost sure convergence using Borel-Cantelli lemma

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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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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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
31 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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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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1answer
31 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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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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7 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
30 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
65 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
17 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
44 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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29 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
16 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
53 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
19 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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29 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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8 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
29 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
24 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 ...