0
votes
1answer
27 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
1
vote
0answers
24 views

How to evaluate this expectation value?

How to evaluate this expected value: $$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le ?,$$ where $\xi_i\overset{ind}{\sim} N(0,1), ...
1
vote
0answers
39 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
1
vote
2answers
55 views

Must every probability distribution over a countable set be discrete?

Intuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as: Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, ...
0
votes
1answer
26 views

Scaling the Lebesgue-Stieltjes integral

Suppose that $F$ is a distribution function. Denote by $\mu_F$ the measure on $\mathbb{R}$ induced by $F$. Suppose that $a>0$. Define a new distribution function $F_a$ by $F_a(x):= F(ax)$, and ...
0
votes
0answers
29 views

The mean of $\mu_{P}(\theta)=\frac{1}{Z}P(x|\theta)$

Consider a parametrized probability measure $P(x|\theta)$, that is for each $\theta\in[a,b]$ it is a valid probability measure on $x$. Denote $f(\theta)$ its mean and $\Sigma(\theta)$ its variance. ...
1
vote
1answer
21 views

Probability distribution “similar” to Gaussian.

Does there exist a distribution A other than Gaussian such that: 1) linear combination of random variable from A is distribution A 2) easy to integrate, for example find entropy Thank you
0
votes
1answer
33 views

Continuity of probability measure

Sorry, I just wanted to know whether I understand this correct. Let $(x_n)$ be an increasing sequence such that $x_n \rightarrow a$, then we have for the probability measure on an arbitrary ...
3
votes
1answer
72 views

Random variable exponentially distributed?

I just want to be sure about this: If I read the phrase ' a random variable is exponentially distributed'( which is often said in probability theory and then it is never explictely stated what $X$ ...
1
vote
0answers
47 views

How to define a probability distribution over a function space?

What is the mathematically rigorous way of defining a probability distribution over some function space e.g. $L^1[0,1]$? Edit: After reading about the basics of measure theory, I realized that the ...
-2
votes
1answer
57 views

Measure Theory and Law of Large numbers

If $X_1,X_2,...$ are non-negative random variables with the same distribution (but the variables are not necessarily independent) and $E[X_1]< \infty $, prove that $$\lim_{n \to ...
0
votes
1answer
22 views

Existence of a measure with given marginals on product space

Let $X_1,...,X_n$, $n\geq 2$ be Polish spaces. I have a given compatible family of probability measures $\{\pi_{ij} \in X_i\times X_j \}$ (here each measure is defined on the space of the form ...
0
votes
1answer
31 views

Show that for $t\in\mathbb{R}$ it is $\int_{-\infty}^{\infty}(G(x+t)-G(x))\, dx=t$ (distribution function)

Let $G$ be the distribution function of a probability measure on $(\mathbb{R},\mathcal{B})$. Show that for $t\in\mathbb{R}$ it is $$ \int_{-\infty}^{\infty}(G(x+t)-G(x))\, dx=t. $$ ...
1
vote
0answers
91 views

Upper Bound on Supremum of Expected Value

Let $\left( \Omega, F, P\right)$ be a probability space, where $P$ is a probability measure on $\mathbb{X} \subseteq \mathbb{R}^n$, so that $P(\mathbb{X}) = 1$. For all integer $i \geq 1$, consider ...
1
vote
1answer
48 views

How do you find the distribution of this sum?

If $X\sim \text{Normal}(\mu=0, \sigma^2), Y\sim \text{Unif}(0,\pi)$, and $X \perp Y$, how do you find the distribution of $Z=X+a\cdot cos(Y)$ for some $a > 0$ ? I've found the distribution ...
0
votes
0answers
55 views

Probability of convergence of a monotone sequence

Let $\left(\Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space, and let $X: \Omega \rightarrow \mathbb{R}^n$ be a random variable. Let $\mathbb{P}^N$ denote the product measure $\mathbb{P} ...
0
votes
1answer
73 views

Is monotonicity condition not required in this short derivation?

For given density functions $p_1(x)$ and $p_0(x)$ ($x\in\mathbb{R}$) the following equation is to be satisfied: $$(1-\epsilon_1)\{P_1[p_1/p_0>c] +cP_0[p_1/p_0\leq c]\}=1$$ where $c\in\mathbb{R}^+$ ...
0
votes
1answer
39 views

For a given random variable X, does there exists another random variable Y in different probability space, X and Y have the same distribution

X (known) is a random variable defined on probability space $(Ω_1 ,F_1 ,P_1 )$ , there exists a random variable Y in another probability space $(Ω_2 ,F_2 ,P_2)$ which has the same distribution as X. ...
4
votes
1answer
147 views

Joint distribution by independent distributions

We have $N$ independent discrete finite random variables (RVs) $X_1,\dots,X_i,\dots,X_N$ where RV $X_i$ has $M_i$ finite number of elements. We are free to choose any distribution $f_i$ for RV $X_i$ ...
3
votes
1answer
353 views

Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $ (\Omega, \mathscr{A}, \mu ) $ ...
4
votes
1answer
44 views

Measurability of integral

Consider a function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ which is continuous in the first argument, measurable in the second. Let $m: \mathcal{B}(\mathbb{R}^m) \rightarrow ...
0
votes
1answer
66 views

Probability of a random variable dependent on a parameter.

Let $X_L$ be a random variable dependent on a parameter $L$, taking only discrete values between $0$ and $+\infty$. Let $\mu L$ be its expectation, where $\mu$ is a costant. Which conditions should I ...
3
votes
2answers
180 views

Expectation of composition of functions with density as R-N derivative

In prior probability courses, I've always seen and used the fact that, for a continuous random variable X and a function $\phi$, $E[\phi(X)]=\int_{ \mathbb{R}}\phi(x) f_X(x)dx,$ but I cannot find a ...
0
votes
0answers
31 views

Conditional on the sum or positive random variables [duplicate]

Let $Y, X_1, . . . , X_n$ be continuous random variables (not necessarily independent) with non negative range, i.e. $P(Y < 0) = 0$ and $P(X_i < 0) = 0$ for $i = 1 \ldots n$, verifying the ...
3
votes
1answer
117 views

Probabilities of uncountable intersection of events

In order to determine a probability for some event $A\in\Omega$, I ended up with $$ \mathbb{P}\left(X_t>f(t),\quad \forall [0,T]\right)≤ \mathbb{P}(A)≤\mathbb{P}\left(X_t≥f(t),\quad \forall ...
0
votes
1answer
180 views

Atomless probability measure

Assume that $m$ is an atomless probability measure on $\mathbb{R}^{d}$. Let $\left( X_{1},\ldots ,X_{d}\right) $ be a random vector with law $m$. Are the marginal cumulative distribution functions of ...
0
votes
0answers
121 views

Sum of Dependent random variables pdf alternative?

I'm attempting to calculate mutual information for two cases and running into some difficulty. Let me describe the setup: $X\sim$ independent and identically distributed symbols $(+1/-1) w/ p=0.5$ ...
3
votes
1answer
199 views

Physical meaning of “probability density”

Is there some way of describing the co-domain of probability density functions? Does it relate in some way to something physically meaningful? I was given that question today - and I was at a loss. ...
1
vote
0answers
47 views

Relation between conditional variances

Let $X,A,Y_1,\dots,Y_n$ be random variables defined on some probability space $(\Omega,\mathcal{F},P)$. Suppose that $Y_1,\dots,Y_n$ are conditionally independent and identically distributed given the ...
3
votes
2answers
64 views

What is the probability of a multidimensional rectangle?

Assume given a probability measure $P$ on $(\mathbb{R}^p,\mathcal{B}_p)$, where $\mathcal{B}_p$ denotes the $p$-dimensional Borel-$\sigma$-algebra. Let $F$ denote the $p$-dimensional CDF for $P$, ...
0
votes
1answer
64 views

A Measure For The Space of Probability Density Functions

Consider the space of all joint probability density functions of two variables. I want to know what the measure is of the portion of this space that is filled by uncorrelated joint pdfs relative to ...
3
votes
3answers
165 views

$E[X]$ finite iff $\sum\limits_{n} P(X>an)$ converges

Show that: $$\sum\limits_{n \in N } P(X>an) < \infty\ \text{for some}\ a > 0 \Rightarrow E[X] < \infty \Rightarrow \sum\limits_{n \in N } P(X>an) < \infty\ \text{for every}\ a > ...
0
votes
0answers
29 views

What is the measure of all probability distributions with finite variance?

I'm in over my head here, but I am wondering about the probability that a distribution has finite variance? (or a finite mean?) By this, I don't mean that there is some set of data, just over the set ...
3
votes
0answers
79 views

A question about the stability of a property of the normal distribution

Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution $P$ on $\mathbb{R}$ with the property that with $X$, $Y$ ...
0
votes
0answers
48 views

Exponential Order Statistics Independence

Are the order statistics from the $n$-sample with $X_i\sim \text{Exp}(\lambda)$ (taking, without loss of generality, $\lambda=1$) $\Delta_{(k)}X=X_{(k)}-X_{(k-1)}$ independent? Can show that for an ...
0
votes
2answers
150 views

Conditional Expectation of Exponential Order Statistic $\text{E}(X_{(2)} \mid X_{(1)}=r_1)$

Having already worked out the distributions of $\Delta_{(2)}X=X_{(2)}-X_{(1)}\sim\text{Exp}(\lambda)$ and of $\Delta_{(1)}X=X_{(1)}\sim\text{Exp}(2\lambda)$ where $X_{(i)}$ are the $i$th order ...
3
votes
1answer
116 views

Exponential Distribution Function

If $X\sim \text{Exp}(X)$ then for all positive $a$ and $b$, $P(X>a+b\mid X>a)=P(X>b).$ So given independent random variables $X \sim \text{Exp}(\lambda)$, $Y \sim\text{Exp}(\mu)$ we would ...
0
votes
0answers
32 views

Gap distribution independence proof

I have a question bout the proof of the independence of gap RVs. Given the independent exponentially distributed random variables $\xi_1$, $\xi_2$ ~ $\text{Exp}(\lambda)$, and a corresponding order ...
6
votes
1answer
85 views

How can a $\sigma$-algebra be “treated” or computed? Example

My question is: I have a random variable $X:\Omega \rightarrow \mathbb{R}$, the $\sigma$-algebra generated by $X$ is: $\sigma(X) := \{X^{-1}(B), B\in \mathcal{B}(\mathbb{R})\}$. But, imagine now that ...
2
votes
0answers
51 views

Two dimensional distribution

Let $F$ be a two variable continous function that satisfy: if $x_1\leq x_2$ and $y_1\leq y_2$ then \begin{equation} F(x_2,y_2)-F(x_2,y_1)-F(x_1,y_2)+F(x_1,y_1)\geq 0. \end{equation} Define ...
3
votes
1answer
306 views

Uniform measure on the rationals between 0 and 1

I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the ...
0
votes
1answer
48 views

(Probably difficult) inequality question for densities under different meaures

Given the measures $(G_0,G_1)$ and $(G^{'}_0,G^{'}_1)$ corresponding to the densities $(g_0,g_1)$ and $(g^{'}_0,g^{'}_1)$ the following inequality $$G_0[g_1/g_0<t^{'}]\geq ...
2
votes
0answers
124 views

What is the total variation measure of the integration of a kernel of signed measures?

Assume given a probability space $(\Omega,\mathcal{F},P)$ and a measurable space $(E,\mathcal{E})$. Let $(\nu_\omega)_{\omega\in\Omega}$ be a family of signed measures on $(E,\mathcal{E})$. Assume ...
4
votes
0answers
212 views

An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.

We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
0
votes
1answer
70 views

Poisson distribution and probability distributions

Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier. Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
1
vote
1answer
45 views

How to understand stationary solution?

How to understand the stationary solution of the stochatic equation: $$X_{n+1}=A_n X_n+B_n$$ And where can I find more information?
0
votes
1answer
51 views

Change of Variables and independent random variables.

Suppose that we have two IID random variables, $X_1, X_2$, carried by a triple $(\Omega,\mathcal{F},P)$. While solving an exercise I ended to a point that I had to see that, $$ \iint\limits_D x_1 ...
3
votes
1answer
183 views

Relationship between two random variables?

What is the relationship between a random variable obeying the subexponential distribution defined here and a random variable $X$ satisfying $P\left(\left|X\right|>t\right)\le\alpha e^{-\beta t}$ ...
5
votes
2answers
564 views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
0
votes
3answers
237 views

measuring distance between probability measures only at the tail

Is there any official (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support? Take, for example, the total ...