Questions about maps from a probability space to a measure space which are measurable.
5
votes
0answers
63 views
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 ...
4
votes
0answers
119 views
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 ...
4
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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:
...
3
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 $X \in Z$ ...
1
vote
0answers
41 views
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?
...
1
vote
0answers
16 views
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 ...
1
vote
0answers
19 views
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 $A>0$, ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
1
vote
0answers
18 views
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 ...
1
vote
0answers
79 views
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
16 views
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 ...
0
votes
0answers
23 views
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 ...
0
votes
0answers
19 views
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
...
0
votes
0answers
26 views
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 ...
0
votes
0answers
30 views
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 ...
0
votes
0answers
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?
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
46 views
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 ...
0
votes
0answers
24 views
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 ...
0
votes
0answers
16 views
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
...
0
votes
0answers
30 views
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 ...
0
votes
0answers
18 views
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 ...
0
votes
0answers
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
$$
E \left|\sum_{i=1}^n ...
0
votes
0answers
17 views
Compansate standard deviation loss
I am not sure if this question will be a little off-topic on this forum, that I will give it a try anyway, since it implies some mathematical background.
By using ...
0
votes
0answers
43 views
How to find the cdf of the minimum of two r.v's?
Let $I=min\{0,W+V-U\}$ where $W,V,U$ are r.v's.
Find the CDF of $I$ ?
0
votes
0answers
36 views
The identity of two parameters derived via conditioning arguments
Suppose I have a random variable $X_1\in\mathbb{R}$ and a random vector $X_2\in\mathbb{R}^d$. Furthermore, there are two measurable functions $f_1$ and $f_2$, and two deterministic vectors $\theta_1, ...
0
votes
0answers
28 views
how to compare correlation between random variables?
Suppose I have a random variable, S(k) for starting date of callable bonds, M(k) for the maturity date of the bonds, and C(k) for the called date of the bonds.
$$S(k) < C(k) < M(k)$$
C(k) is ...
0
votes
0answers
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 ...
0
votes
0answers
21 views
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 ...
0
votes
0answers
98 views
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 ...
0
votes
0answers
62 views
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 ...
0
votes
0answers
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 + ...
0
votes
0answers
50 views
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 ...
0
votes
0answers
58 views
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 $ ?
0
votes
0answers
48 views
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| > ...
0
votes
0answers
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 ...
