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 ...

learn more… | top users | synonyms (1)

3
votes
1answer
44 views

Let $(X_i)$ be i.i.d. exponential, is the set $\{X_1,X_2,\ldots\}$ almost surely dense in $(0,\infty)$?

To be clear, I'm asking if the range of the random sequence $(X_i)$ is dense in $(0,\infty)$ a.s. I thinks the answer is yes, because for any $0<a<b<\infty$, we have $$P(X_1 \notin (a,b), ...
2
votes
1answer
84 views

Conditioning a conditional probability to a sigma algebra

Suppose I have two random variables, $X$ and $Y$, defined on the space $(\Omega,\mathcal{F},P_1)$ which can both take the values $0,1,\ldots,N$. Suppose further I want to define the probability of ...
0
votes
1answer
15 views

Convergence in mean square and almost surely

Given the sample space [0,1] and the uniform probability measure P(.), random variables $(X_n)_{n\geq1}$ are defined by How do I $X_n$ converges almost surely as n tends to infinity and also in ...
1
vote
1answer
15 views

Link between conditional characteristic function and conditional density

Let $X$ and $Y$ be random variables (real-valued). I define $$E[e^{i\theta X}\mid\sigma(Y)] =: g(Y,\theta)$$ Suppose that $g(Y,\theta) = e^{i\theta Y}e^{-\frac{1}{2}\theta^2}$. Can I then say that ...
0
votes
1answer
16 views

Proving This Theorem on Independence

I'm trying to find a good resource for proving the following theorem, stated in Shreve's "Stochastic Calculus for Finance II," p. 73: Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let ...
0
votes
0answers
12 views

Scheduling in Wireless Networks with Rayleigh-Fading Interference Simulation Results

I was going through the paper - Scheduling in Wireless Networks with Rayleigh-Fading Interference. Link to PDF: https://people.mpi-inf.mpg.de/~mhoefer/08-0x/Dams14RayleighJ.pdf How to get their ...
1
vote
0answers
22 views

Variance of Conditional Expectation From First Principles

Let $X : \Omega \to S \subset \mathbb{R}$ and $Y : \Omega \to T \subset \mathbb{R}$ be random variables on $(\Omega, \mathcal{F}, P)$. Form the conditional expectation $$ E(Y \mid X) := E(Y \mid ...
1
vote
1answer
19 views

Characteristic function of discret random variable

I try to show the following: Suppose $(X_n),n\geq1$ is a sequence of random variables with uniform distribution on $\{1/n,\dots,n/n \}$. Show that $(X_n)$ converges in distribution to a random ...
0
votes
0answers
12 views

Spectral Density of an ARMA process.

For an upcoming Stochastic Processes exam, we have had a sudden brief email about Spectral Density as the lecturer had forgotten to mention it in classes. He states, For an ARMA process with ...
-1
votes
1answer
17 views

Display the quotient between the value of an histograms

i'm triying to display the quotient of two histograms which have the same borders. Is it possible to do is with Matlab ? Thanks a lot !
1
vote
1answer
50 views

Covariance of stochastic integral

I have a big problem with such a task: Calculate $\text{Cov} \, (X_t,X_r)$ where $X_t=\int_0^ts^3W_s \, dW_s$, $t \ge 0$. I've tried to do this in this way: setting up $t \le r$ $$\text{Cov} \, ...
-1
votes
0answers
20 views

Converges a.s. given convergence in distribution

Assume that we have a sequence of R.V. $X_n$ for which the following holds for some $\mu$ $$\sqrt{n}(X_n-\mu)\rightarrow N(0,1) $$ as $n\rightarrow \infty$ where the convergence is in distribution. ...
0
votes
1answer
38 views

Poisson distribution with rationals

We want to construct an infinitely divisible random variable $X$ supported on the rational numbers as follows: Let $N$ be a Poisson random variable with some parameter $\lambda >0$ Let $R$ be any ...
0
votes
1answer
30 views

Strong Law of Large Numbers and Convergence a.e.

Let $(\Omega,\mathcal F,P)$ be a probability space. A sequence of r.v.'s $X_n$ converges a.e. to $X$ if and only if there exists a null set $N$, such that: $\forall \omega\in\Omega\setminus ...
0
votes
1answer
23 views

Continuous conditional distribution

This is about continuous conditional probability distributions. Why is it that they are allowed to take on single values, while this is a no-no with non conditional continuous distributions(to my ...
0
votes
1answer
15 views

Questions about conditional expectation based on Reed-Frost model.

Firstly, I know the basic theory of conditional expectation given $\sigma$-algebra. I'm reading a book on stochastic epidemic models. In Reed-Frost model there's a sequence of r.v's $(X_{t})$ where ...
0
votes
0answers
15 views

How many people should I ask if a statement (A) is true if the same can be inferred by asking two other statements (X and Y implies A)?

I am asking a number of participants if they believe a given statement is valid. I have a number of such statements, some of which can be inferred. In the made up example below, ...
1
vote
0answers
26 views

subaddivity of VaR

It is known that the VaR (Value at risk) doesn't fulfill subadditivity, i.e. $VaR(X)+VaR(Y) \le VaR(X+Y)$. But for elliptical distributions subadditivity is true. Questions: (1) Which ...
0
votes
0answers
27 views

Probability function

Can anyone give me the Probability Density function of a continuous random variable without given interval, that the interval being $(-\infty,\infty)$ and the function defined as $f(x) = e^{-x} \cdot ...
0
votes
1answer
48 views

Basics of probability theory - fifth graders homework

If you write some four-digit numbers using the digits $1, 2, 3, 4,$ place them in the bag, what is the probability of getting the number, which is $4$ in the ones digit(first from the right side)?
1
vote
0answers
17 views

Counterexamples of Cumulative Distribution Function ( multidimensional )

For simplicity, we consider 2-dimensional. We consider the function $F:\mathbb{R}^2\to\mathbb{R}$. Let $F$ satisfies: $0\leq F(x_1,x_2)\leq1$ for $(x_1,x_2)\in\mathbb{R}^2$ ...
0
votes
0answers
20 views

Transition functions induced by Markov processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and denote by $(X_t,\mathcal{F}_t)_{t\geq 0}$ a time-continuous Markov process with values in $(E,\mathcal{E})$. For $s<t\in ...
1
vote
0answers
51 views
+50

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
0
votes
1answer
14 views

Using probability or moment generating functions to find the distribution when given the distribution with random parameters

Problem: Given $X|M=m \sim $ Poi(m) with $M\sim $ Exp(1). Find the unconditional distribution of X where M is the random parameter. I want to solve this using P.G.F $g(t)$ and/or M.G.F $\psi(t)$. ...
0
votes
1answer
32 views

Use the delta method to find the distribution of $Z_n$

Let $\overline{X}_n=\overline{X}$ the sample mean such that $\sqrt{n}\overline{X}_n\rightarrow^D N(0,1)$ where $\rightarrow^D$ means converge in distribution. Use the delta method to find the ...
0
votes
1answer
24 views

Simple inequality of tails of random variables

Let $a>0$, $X$ some random variable and $\mathbb{E}[X]=\mu<t$. I was trying to prove the following simple inequality: $$ \textrm{Pr}\left[|X-t|\geq a\right]\leq\textrm{Pr}\left[|X-\mu|\geq ...
0
votes
2answers
26 views

Relative entropy (KL divergence) of sum of random variables

Suppose we have two independent random variables, $X$ and $Y$, with different probability distributions. What is the relative entropy between pdf of $X$ and $X+Y$, i.e. $$D(P_X||P_{X+Y})$$ assume all ...
3
votes
1answer
43 views

Is it almost impossible for the values of continuous random variables to lie on a plane?

Let $X: \Omega \to \mathbb{R}^d$ be a random variable with density $f$ (the pushforward measure on $\mathbb{R}^d$ is absolutely continuous). Let $x_1, \dots, x_{d}$ be the set of values of $d$ i.i.d. ...
0
votes
0answers
20 views

Convergence of $\prod\limits_{t\in\mathbb{N}}P(|X_t|<\lambda^t)$, where $\lambda>1$

Let $\{X_t\}_{t\in\mathbb{N}}$ be an independent sequence of continuous random variables on the real line. Let $\lambda>1$. I am interested in the quantity $$ P(\forall t\in\mathbb{N},\,|X_t| \le ...
1
vote
1answer
26 views

Drawing Random Variables

What does it actually mean when we draw a number from a given distribution? To elaborate on what exactly I'm trying to ask: if I generate a vector of numbers from the standard normal distribution on ...
0
votes
1answer
28 views

Find the PDFs of X+Y [closed]

If $X$, $Y$ have joint PDF, $$f(x,y) = \begin{cases} 1 & \text{if }~0 \le y \le 1,~2y\le x \le 2 \\[1ex] 0 &\text{if elsewhere} \end{cases}$$ Find the PDF of $X + Y$. Hi, I am stuck on this ...
2
votes
1answer
47 views

application of Birkhoff Ergodic Theorem

Let $(\Omega, \mathcal{F} , P)$ be a probability space and let $T$ and $S$ be ergodic, measure-preserving transformations of $(\Omega, \mathcal{F} , P)$. Let $X : \Omega → \mathbb{R}$ be a bounded ...
2
votes
1answer
91 views

Independence of RVs conditioned on an event

Assume that we know that $X_1 \in R^n$ and $X_2 \in R^n$ are independent conditioned on $Y=y$, that is, for any $E_1 \subset R^n$ and $E_2 \subset R^n$, $P(X_1 \in E_1, X_2 \in E_2 \vert Y=y) = ...
-1
votes
1answer
47 views

A question of determining when the entropy is maximum.

Y ={ 1, 2,...,r} We are given that X is the set of two sided sequences with entries from Y and T is the two sided shift on X, and m is a T invariant probability measure on X. If $p_i = m(\{x ...
2
votes
2answers
62 views

Discounted price process in Black-Scholes model is a martingale with respect to Q.

I have been presented a proof that the discounted price process in the Black and Scholes formula is a martingale, but there is something important omitted, and I am not able to fill in the gap. I will ...
0
votes
0answers
40 views

Feller property of Ito diffusion

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded and continuous, ...
0
votes
0answers
9 views

What does fixed regressor say about our linearity condition?

The linearity condition states that $y_i=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for those ...
0
votes
1answer
27 views

Do these integrals converge to 0?

Assume that you have a probability space $(\Omega,\mathcal{F},P)$. And you have a positive random variable $Z$, with $E[Z] =1$. You can then define a new probability space $(\Omega,\mathcal{F},P')$, ...
0
votes
1answer
46 views

Bivariate normal distribution in polar coordinates with unknown correlation between the variables.

Consider the problem of finding the dist. of $\theta$ after changing a two dimensional normal distribution to polar coordinates where both variables are standard. Using transformation theorem I get; ...
2
votes
1answer
15 views

Definition of Cylindrical Brownian Motion and Spatial Correlation

From Gawarecki and Mandrekar, Stochastic Differential Equations in Infinite Dimensions: We call a family $\{ W_t \}_{t\geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, ...
0
votes
1answer
18 views

What is the probability distribution of the following random variable?

Let $A^n$ and $B^n$ be independent random variables taking values in $\{0, 1\}^n$. Let $Y^n = A^n + B^n$ (Hence, taking values in $\{0, 1, 2\}^n$). How can we express the distribution of $Y^n$ in the ...
0
votes
1answer
21 views

Calculating convolution of binomial distribution using moment generating function

I have two independent random variables $X_{1}, X_{2}$ on the same probability space. $X_{1}$ is bin bin(n, p) and $X_{2}$ ís bin (m, p) with n, m natural numbers and p in the interval [0,1]. I need ...
0
votes
0answers
16 views

Relation between Kolmogorov Zero-One Law and Random Graphs Zero-One Laws?

I know of Zero One laws for Random graphs (such as those concerning monotonic or first-order-logic properties). I also know about Kolmogorov's zero one law for tail sigma algebras. Apart from the ...
1
vote
0answers
16 views

What is the chance that two random binary random variables are independent?

Consider the space of probability distributions on 4 letters. Now all the probability distributions on four letters do not represent a distribution of two independent binary random variables. But if ...
0
votes
1answer
45 views

non negative super martingale

Let $(X_n)_{n\geq0}$ be a non-negative supermatingale and $T = \inf\{n \geq 0 : X_n = 0\}$. Show that on the event $\{T < \infty\}$, $X_{T+n} = 0$ for all $n \geq 0$ a.s. My approach: $0 \leq ...
1
vote
1answer
45 views

Show $\mathbb{E}(\sum_{i=1}^n(X_i-(\frac{1}{n}\sum_{i=1}^{n}X_i))^2)=(n-1)\mathbb{V}(X)$

Let $X_1, X_2, \ldots$ be a sequence of i.i.d random variables with finite variance and $M_n=\frac{1}{n}\sum_{i=1}^{n}X_i$. We need to show that ...
1
vote
1answer
22 views

Find the cdf $F_{X,Y}(u,v)$ if the pdf is given by $f_{X,Y}(x,y) = 6x$ for $0\leq x \leq 1$ and $0 \leq y \leq 1-x$

Find the cdf $F_{X,Y}(u,v)$ if the pdf is given by $$f_{X,Y}(x,y) = 6x$$ for $0\leq x \leq 1$ and $0 \leq y \leq 1-x$ I have the solution to this, but I don't understand it completely. Can some one ...
0
votes
0answers
17 views

What are the differences in linearity in Non-stochastic and Stochastic Regression?

I have been confused with the differences between stochastic and non-stochastic explanatory variables for a while. I was able to write down some of my understanding and seek approval or comments about ...
0
votes
1answer
23 views

Conceptual/Notational question on conditional distributions and “given”

So in the book I'm reading, I see the notations $f(x|\theta)$ being used to refer to population distributions, dependent on $\theta$ which are in a family. The author explains this as a notational ...
1
vote
0answers
25 views

Feedback of proof that $\lim \inf A_n \subseteq \lim \sup A_n$

I just wrote down the proof of the following easy proposition, and I was wondering about both the content (I would like to know if it is error-free), and the form of it. Proposition: $\lim \inf ...