# Tagged Questions

Questions about maps from a probability space to a measure space which are measurable.

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### What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
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### Existence of iid random variables

In probability theory we often used the existence of a sequence $(X_n)_n$ of independent and identically distributed random variables. This was already discussed here. One of the answers says: As ...
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243 views

### Characteristic functions of random variables (Poisson, Gamma, etc.)

My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is ...
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30 views

### Random variables related through nonlinear system of equations

Lets assume two groups of random variables X and Y (the dimensionality of them is not important). I know probability distribution of X, but not of Y. I also know that Y is a function of X and they are ...
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58 views

### Relation between factor graph and conditional probability distribution

First, I'm from computer science. I don't know how to say this problem in a mathematical way. So please bear with me. The question Let say I have a factor graph illustrated in the figure. The ...
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49 views

### Convergence In $L^{1}$ in the Strong Law of Large Numbers

I'm trying to prove that if $(X_n)_{n\geq 1}$ is uniformly integrable, then $X_n$ almost surely converging to $X$ implies $X_n$ converges to $X$ in $L^{1}$. How is this done? Generally speaking: ...
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59 views

### Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $\mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I)$ is a random variable? The kronecker product ...
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18 views

### Fast fourier transforms of random binary data

I am a physicist who is trying to make sense of FFTs and binary data. Say I have a series of random binary data, which is measured with a repetition rate of 400Hz (interval time of 0.0025s). I have a ...
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28 views

### convergence of discrete random variables with finite entropy

Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ ...
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### About Strict Stationary of AR(1) Sequence

The usual Auto regressive process considers the time t from negative infinity and positive infinity, but what if we restrict our time to strict positive space, do we still have our stationary result? ...
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### Strong Law of Large Numbers a.s. sense implies $L^1$ sense?

I have to show that the strong law of large numbers in the almost sure sense implies the strong law of large numbers in the $L^1$ sense. I'm not sure what's being asked, can anyone give me a hint or ...
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### Generating constrained random numbers

I need to find a way to generate random vectors $v \in \mathbb{R}^{n_v}$ and $w \in \mathcal{R}^{n_w}$ such that they satisfy the condition $$R_1 v + R_2 w = c,$$ where \$R_1 \in \mathbb{R}^{n_c \times ...
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39 views

### Showing a certain process has $\limsup X_t$ bounded almost surely.

This question has been solved. I'm working on a problem where I need to show $$\limsup_{t \rightarrow \infty} X_t \leq \sqrt{c}\quad \text{a.s.}$$ where $X_t$, $t \geq 0$ is a stochastic process ...
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48 views

### Property of Sum of Random Variables

Let $\left(X_n\right)_{n\geq 1}$ be a sequence of i.i.d. real random variables, with $\mathbb E(X_1)=0$, $\operatorname{var}(X_1)=1$. Let $S_n=X_1+\cdots+X_n$. Prove that for any $A>0$, ...
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53 views

### Definition of $x\left<Y\right>$ notation in probability theory

I am working on the basics of probability theory in Koller's Probabilistic Graphical Models - Principles and Techniques. Unfortunately I am having trouble understanding a formal definition (possibly ...
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65 views

### A basic question on limit

Why for a continuous random variable $X$ there exists a $\delta > 0$ such that for all $x$ in $[c, c+\delta]$ the following is true: $$P(c < X \leq x) < \epsilon$$ for any given \$\epsilon ...
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163 views

### Random Variable on a Sphere

Not sure where to start with this problem: For any $d\geq 1$, we admit that there is only one probability measure $\mu$ on $\mathcal S_d$, (the $(d-1)-th$ dimensional sphere embedded in \$\mathbb ...
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### Estimating the likelihood of independence of two discrete variables using the co-occurrence count matrix.

I have some data about users from different regions visiting different directories of some website. Aggregating that data I get the co-occurrence frequency matrix (for regions and directories). Now I ...
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### How to calculate the highest/smallest possible value of the variance of two random variables mean?

Two random variables $X$ and $Y$ have a common expected value $E(s)$ and a common variance $Var(s)$. What's the highest possible value of the variance of their mean, $var ((x+y)/2)$? What's the ...
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51 views

### Measuring a mean variance for some number of objects observed per trial for multiple trials

I'm running a bunch of trials, $T$, and the outcome of each trial is some number of objects $k_i$ for $i = [1, T]$. I would like to say something about the average "spread" in terms of the number of ...
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93 views

### Independent Exponentially Distributed Random Variables - Athletes Problem??

Q) At a javalin competition two athletes (1 & 2) are competing against each other. Each has one attempt to throw the javalin. Assume the acheived distance of a throw ($L$1 & $L2$) [note these ...
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83 views

### circular complex random vectors

Is a vector which components are circular (aka proper) complex random variables also circular complex? Below I summarized my attempt to solve this problem. I think the answer is no, but the ...
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### how to write the joint density fuction of two variables that obey lognormal distribution

Suppose $U_t$ is a random variable subject to $\operatorname{Lognormal}(x_1, z_1^2)$ distribution. $V_t$ is a random variable subject to $\operatorname{Lognormal}(x_2,z_2^2)$ distribution. Suppose ...
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### P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined \$b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
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### How to find the average in the given labels?

I generated 11 random variables follow the Poisson distribution. I used lambda equal to 5. The data that I got is following : 0.00673794699908547 0.0336897349954273 0.0842243374885683 ...
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### Covariance between sample variance and sample sum of squares

I am trying to find the cov(A,B), where A is the sample variance, and B is the sample sum of squares. I am new here, and don't know yet how to enter the formuale in the question box, but I think they ...
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### Homework Help. Probability Density Functions.

$X$ is $N(10,1)$. Find $f(x|(x-10)^2 < 4)$ This is a homework question. I can only figure out that X is normally distributed with mean 10 and variance 1. Can you please explain what is meant to ...
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31 views

### a sequence of random variables that converge to a constant c in probability but fail to converge to c with probability 1?

Any example that a sequence of random variables that converge to a constant c in probability but fail to converge to c with probability 1?
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26 views

### Skewness of a sum with a positive summand

Let $X$ and $Z$ be two random variables with finite third moment, and let $Z>0$. Is it true that the skewness of $X+Z$ is greater or equal than that of $X$? Such a relation clearly holds for the ...
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17 views

### Is $f_{\Theta|Z}(\theta|z)$ Gaussian when $Z = \theta^3 + V$, and given that $\Theta$ and $V$ are Gaussian?

$\Theta$ and $V$ are zero mean Gaussian random variables with variances $\sigma_\Theta^2$ and $\sigma_V^2$. A third random variable $Z$ is defined as: $$Z = \Theta^3 + V$$ Is ...
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17 views

### Generate Constrained Vector of Random Numbers?

I'm having trouble creating a random vector $\vec{V}$ starting with a standard 0:1 randon number generator subject to the following set of constraints: (given parameters $D$, $L$, and $\theta$) The ...
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### How to prove that $Y=\ln(X)$ approximately Normal when $X$ is a Normal random variable with $\mu\gg\sigma$

I wanted to prove that PDF of $Y=\ln(X)$ tends to a Normal distribution with $\mathcal{N}(\ln(\mu_{x}),\sigma^{2}_{y})$ when $X\sim\mathcal{N}(\mu_{x},\sigma^{2}_{x})$. It is also important to note ...
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### Calculation of the error function.

I have the next two signals: $X(t)$ and $G(t)$ and a random process $Y(t)=G(t)X(t)$ where $X(t)$ and $G(t)$ are wide sense stationary with expectation values: $E(X)=0, E(G)=1$. Now, it's also given ...
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### How to test whether there is an association between two data fields by testing a hypothesis?

The table below cross classifies Education by Employment Confidence and is based on a sample 1363 randomly selected adult respondents in China. Highest degree　　　　　　　　　Employment Confidence　　　　Total ...
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### Using an appropriate hypothesis to test whether two means are different

Manager examined potential differences between two models of bicycles. The mean life of the bicycles is of primary concern. The followings table provides the available date which measured in ...
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### is there a Kalman filter for distribution function?

The standard Kalman filter uses a series of measurements observed over time, to decomposite the signal and noise. However, when I'm modeling the distribution (pdf or cdf) of a variant, is there a ...
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31 views

### Inequality for expected values

Let $x=(x_1, \ldots, x_n)$ be real valued vector. Let $\pi(\cdot)$ be a permutation on the set $\{1, \ldots, n\}$ with a uniform distribution. Prove the following inequality  E \left|\sum_{i=1}^n ...
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17 views

### Compansate standard deviation loss

I am not sure if this question will be a little off-topic on this forum, that I will give it a try anyway, since it implies some mathematical background. By using ...
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43 views

### How to find the cdf of the minimum of two r.v's?

Let $I=min\{0,W+V-U\}$ where $W,V,U$ are r.v's. Find the CDF of $I$ ?
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### The identity of two parameters derived via conditioning arguments

Suppose I have a random variable $X_1\in\mathbb{R}$ and a random vector $X_2\in\mathbb{R}^d$. Furthermore, there are two measurable functions $f_1$ and $f_2$, and two deterministic vectors \$\theta_1, ...
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28 views

### how to compare correlation between random variables?

Suppose I have a random variable, S(k) for starting date of callable bonds, M(k) for the maturity date of the bonds, and C(k) for the called date of the bonds. $$S(k) < C(k) < M(k)$$ C(k) is ...
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32 views

### Correlation Coefficient dealing with discretely distributed variables

I'm a bit stuck on this practice problem I have for my HS business stats class. I'd appreciate any help to get the solutions. Thank you. Exercise #22: Let X and Y be discretely distributed random ...
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### Combining function estimates

I have two piecewise linear estimates for two different realisations of the same random variable. What are some techniques that I could use to combine these function estimates into a single ...
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### what is the pdf of the product of two independent RVs for Normal and chi-square distributed RVs?

what is the pdf of the product of two independent random variables X and Y, if X and Y are independent? X is normal distributed and Y is chi-square distributed. Z = XY if $X$ has normal distribution ...
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### Transformation of Discrete Random Variable

Suppose i have a discrete random variable A such that: $p(A=-1) = 3/4$ $p(A=0) = 1/8$ $p(A=1) = 1/8$ Now, i create a random variable $B = |A|$ and so $p(B=0)= 1/8$ $p(B=1)= 7/8$ I want to ...
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94 views

### conditional expectation of normal random variables

I am not the best when it comes to conditional expectation and I am currently going over an economic/finance theory paper and they have the following statement: $V_D = \delta_D + d_D$ x \$\delta_F + ...
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### Min and Max Correlation values

i want to know whats is the max value that the correlation rho can take : $\mbox{Rho}(u_1,u_2) = \mbox{Cov}(u_1,u_2)/(\sigma_1*\sigma_2)$ where $u_1$ is a uniform random Variable and $u_2$ is an ...
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### Convergence in series of expectation

If $X_i$ are independent and the series of $\sum X_i$ is convergent. $\sum X_i = Y$ , does it imply $\sum EX_i = EY$ ?
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### How distribution function behaves when $b\to\infty$

Let $X$ be a r.v. with dist func $F$ and let $a < b$ .Find and sketch the distribution function of z=\begin{cases} x & \text{if } |x| \leq b \\ 0 & \text{if } |x| > ...
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194 views

### Integral of a gaussian with random variance

Assuming: $$X(x,\mu)=\frac{1}{\sqrt{(2\pi)\sigma^2}} \exp[-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}]$$ the integral of $X(x,\mu)$ from $-\infty$ to $+\infty$ is: S=\int_{-\infty}^{+\infty}dx ...
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