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

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What is the distribution of the subtract of two random variables?

Definition) A stochastic process $\{X(t), t \geqslant 0\}$ is said to be Brownian motion process with drift coefficient $\mu$ and variance parameter $\sigma^2$, if it satisfies that $X(0)=0$. $\{X(t)...
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Can continuous random variables ever have positive probability on a single point?

From a textbook: Continuous random variables can lead to confusion. First, note that if $X$ is continuous then $\mathbb{P}(X = x) = 0$ for every $x$. But then later: Let $F$ be the CDF for a ...
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Are these definitions of a continuous random variable equivalent?

In a textbook I'm reading: A random variable $X$ is continuous if there exists a function $f_X$ such that $f_X(x) \ge 0$ for all $x$, $\int_\infty^\infty f_X(x) dx = 1$ and for every $a \le b$, ...
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Convolution: Give a proof that $f_T(t)=\int_{-\infty}^{\infty}f_X(x)f_Y(t-x)dx$ where $f_T(t)$ is the PDF of random variable T

Here is the question: Let $X$ and $Y$ be independent, continuous r.v.s with PDFs $f_X$ and $f_Y$ respectively, and let $T=X+Y$. Find the join PDF of $T$ and $X$, and use this to give a proof that $...
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37 views

Change of Uniform Continuous Variable

Let $X$ be a $U(-1, 1)$ random variable, we define $Y = X^4$. Calculate the correlation coefficient between both variables. Are they uncorrelated? PS. I don't know how to use MatJax equations, I'm ...
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Expected values of Cereal box - Linearity of expectation puzzle [duplicate]

A toy is randomly put in a given Cereal box as a promotional gift. There can be N different types of toys and each one can be of any type N (IID). (a) Find the expected number of cereal box one has to ...
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16 views

Do characteristic functions characterize the independence of random variables? [Solved] [duplicate]

It is well known that the probability density function characterizes the independence of random variables in the following sense. $$X,Y \quad\text{independent}\iff f(x,y)=f_x(x)f_y(y)$$ where $f$ is ...
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Why does the mean centered autocorrelation have a slope of -1?

I'm fundamentally not understanding something about the autocorrelation function (as defined by numpy.correlate). Let's say I create a bunch of random signals $s_1, ...
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78 views

Interpretation of correlation (coefficient)

In an discussion we were confronted with a very special opinion about correlation in respect of financial assets. The widely used correlation coefficient is used here to give an idea about how ...
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67 views

What is the pdf of sum of log-normal and normal distribution?

The question goes like this: $Z = X+Y$; where $X$ is Log-normal Random variable with parameters - $\mu = 0 \quad \sigma^2= 1$, $Y$ is Gaussian Random variable with $\mu= 0\quad \sigma^2= 1$ What is ...
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Notation: should Markov chains steps be noted by uppercase or lowercase letters?

I'm reading the chapter about perfect sampling of the "Monte Carlo Statistical Methods" by Robert and Casella, 2004. I've got an issue about notation, when they talk about random mappings, they say $$...
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inverse Mapping in Transformation of a random variable

I have a question concerning the the inverse mapping in the image . text extracted from Casella Statistical inference $g^{-1}(A) = \{ x \in \chi : g(x) \in A\}$ I know the idea that they want to ...
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51 views

How to generate correlated random numbers with specific distributions?

After read the answers of some similar questions on this site, e.g., Generate Correlated Normal Random Variables Generate correlated random numbers precisely I wonder whether such approaches can ...
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Density function of a gaussian vector

Let $X=(X^{(1)}, ..., X^{(n)})$ be a random gaussian vector with mean $m\in\mathbb{R}^{n}$ and covariance matrix $\Sigma$ with $\text{det}(\Sigma)\neq 0$. Prove that the distribution of $X$ is ...
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1answer
35 views

is Cov(X) and Var(X) same? when X is random vector

i'm studying with hogg. introduction to mathematical statistics. and i learned about random vector but i wonder whether Cov(X) and Var(X) is same or not. as intuitive thinking , if X is a random ...
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22 views

Independent coordinates of a gaussian vector

Let $(X^{(1)}, \ldots, X^{(n)})$ be a gaussian random vector. For $i\neq j$, Prove that $X^{(i)}$ and $X^{(j)}$ are independent $\iff \text{Cov}(X^{(i)},X^{(j)})=0$. I'm trying to work with the ...
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Probability of drawing in the right order and having the second draw be drawn before a fixed step

Suppose I am drawing objects uniformly at random, and I continue drawing without replacement until all objects are listed. So the object I draw at the first step is listed in the first place, the ...
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23 views

Generaling dependent random variables

You wish to generate three standard normals $X, Y$ and $Z$ with correlation matrix given by $$R =\begin{pmatrix} 1.0 & 0.2 & 0.2 \\ 0.2 & 1.0 & 0.2 \\ 0.2 & 0.2 & 1.0 \end{...
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51 views

Finding the Expected Value with a Random Constant

Suppose $X$ is a continuous random variable with PDF: $$\begin{cases} e^{-(x-c)}\ \ \text{when }x > c \\ 0\ \quad \quad\text{when}\ x \leq c \end{cases}$$ a. Find $\mathbb{E}(X)$ b. Find $\...
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42 views

Word Problem: Probability of Y books Fitting in Book Case

Problem: You have $4600$ cm of book case. The thickness of the books are independently distributed with $X \sim N(1.8$ cm$,0.7^2)$. Approximately determine what the probability of ...
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Conditional expectation and set times random variable??

On page 62, what in the world is the meaning of equation (5.2)? $\mathcal{F}_t$ is a $\sigma$-algebra, so $Z_t \in \mathcal{F}_t$ is a set. $X_u$ is a random variable, so what is $Z_t X_u$?
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Derivation of the Negative Hypergeometric distribution's expected value using indicator variables

I'm trying to understand how to derive the Negative Hypergeometric's expected value using indicator variables. Note, in the problem below, we are only interested in the expected value before the first ...
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1answer
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Asymptotic inner product of correlated random vectors

Suppose $\mathbf{x}$ and $\mathbf{y}$ are N-dimensional non-white complex random vectors independent of each other i.e., covariance matrices $\mathbf{C_{xx}}\neq\mathbf{I}$, $\mathbf{C_{yy}}\neq\...
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58 views

Expected value of X and Y for a given problem

A couple decides to have children until they get a girl, but they agree to stop with a maximum of 5 children even if they haven't gotten a girl. If X and Y denote the number of children and number of ...
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26 views

Calculating the magnitude of random numbers from normal distribution

Statement: Given an array of 80 random numbers, normally distributed between 0 and 1, we can expect that the numbers are all of similar magnitude, on the order of $80^{-1/2} \approx 0.1$. Question: ...
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Calculation of time autocorrelation

Given S(f) where it's the PSD of a random process X(t), required to calculate time autocorrelation function of the random process X(t) using the following sample function X1(t) = cos(wc t + π/4) Does ...
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1answer
30 views

What is the domain of a function of random variables?

Consider a random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ $X:\Omega \rightarrow \mathcal{X}\subset \mathbb{R}$. Suppose $X$ has range (or image) $\mathcal{I}\subset \...
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34 views

Proof of discrete probability monotone convergence

I am trying to show that for a sequence of random variables defined on a sample space $\Omega$ $$0\leq X_1(\omega)\leq X_2(\omega \leq ......\leq X_{n}(\omega)...$$ for all $\omega\in\Omega$, with $...
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1answer
41 views

Minimizing MMSE over positive random variables

Let X be a random variable with a finite second moment. We know that argmin E(X-Y)^2 = E(X|g), Where the minimum is taken over all g-measurable random variables Y. How can I find argmin E(X-Y)^2 ...
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1answer
22 views

Strong Markov Property for Markov Chains - Statement Verification

I suspect that my handwritten lecture notes for the Strong Markov Property are wrong. I'd appreciate corrections to them. We first define the following: A random variable $\tau$ is called a ...
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36 views

Terminology of “Random variable”

A random variable $X$ is a measurable function $X : \Omega \rightarrow E $ where $\Omega$ and $E$ are measurable sets. So, as far as I can see from this definition, random variables are just ...
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The definition of random sequence

Suppose that I ask you to tell me four integers between $0$ and $10$ randomly. You tell your numbers, for example $\{3,7,2, 5\}$. However I don't trust you about your numbers being random, hence I ...
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Why is $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X > k)$ true?

I'm working through a problem regarding expected values in Markov chains, and at some point it says: Recall from probability that if $X$ is a positive integer valued random variable, then $\mathbb{...
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Determining distribution and therefrom probability

The problem is as follows: Assume that $V_1$ and $V_2$ are independent random variables with $V_1 \sim \chi^2(5), V_2\sim\chi^2(9)$. Find the value of $b$ such that: $$P[\frac {V_1}{V_1 + V_2} \lt b] ...
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Functions of random variables result, where does it come from

I have learned that if one has two random variables, say $X$ and $Y$ and if $Y=g(x)$, then we have that density of r.v. $Y$ is: $$f_Y(y) = f_X(g^{-1}(y))\left| \frac{d(g^{-1}(y))}{dx}\right|$$ This ...
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Correlation Coefficient of Random Variables

Question: My work for parts a and b: Now I'm stuck with part c and don't know where to go or how to get the answer from parts a and b. any help?
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Two random variables with same moments

Reading http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter10.pdf pages 368-370. it states "if we delete the hypothesis that have finite range in the above theorem, ...
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3 Red cards and 2 Yellow. Calculate the expected value and Variance

So this is how it goes. In a pack of cards there're 3 red cards and 2 yellow cards. In each step we take out cards one by one (without returning) until we firstly get one of each color. Find out the ...
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What model should I use for judging a dimension given only composed data with another?

I am attempting to upgrade a modeling system using a limited type of statistical information, but with the sample covering the entire system. The problem is how to use the additional information in ...
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1answer
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Independence from factors implies independence from the product?

Edited: If $X$ is independent from $Y$ and $Z$, is it true that $X$ is independent from $YZ$?
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Sigma algebra generated by the stopped process.

Let $(X_n)_{n \geq 0}$ be a sequence of random variables. Let $\mathcal{F}_n = \sigma (X_0, \dots, X_n)$ be a filtration and $T$ is a $(\mathcal{F}_n)_{n\geq 0}$-stopping time. I want to understand ...
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1answer
45 views

Related problem to covering a circle with random arcs

I have a problem setup wherein we have (the following are all integers) a sequence of length $G$, and $N$ reads of length $L$. I'm interested in the problem where we consider the sequence to be ...
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1answer
50 views

Most likely order of independent normal random events

The problem I have is, given $n$ independent normal distributions describing the times that $n$ random events occur at, what is the most likely order that they will occur in? This questions follows ...
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$Y$ can only take on $\{−1, 0, 1\}$. The expected value of $Y$ is $0$ and its variance is $1/2$. Find the probability distribution of $Y$.

How would one approach this question? A random variable Y can only take values in $\{−1, 0, 1\}$. The expected value of $Y$ is $0$ and its variance is $1/2$. Find the probability distribution of $Y$. ...
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1answer
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Two exponentially distributed random variables w/ different intensity. Which is more probable to take?

Let's say I have two types of light bulbs, A which has $E(A)=100$ hours of lifetime, and B which has $E(B)=200$. I have three of type A and one of type B. I randomly use one of the four, and after 200 ...
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Probability of a right angled triangle with sides a+b=200 having hypotenuse ≥ 160

QUESTION: A $200\, cm$ long staff breaks in two at a random point. The two parts becomes the right sides of a right angled triangle. What is the probability of the hypotenuse being at least $160\,cm$? ...
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21 views

non square transformation of random variables

Let $x_0$ and $w_0$ be independent random variables and let $x_1$ be related to them by $x_1 = f(x_0, w_0)$. I want to find the joint density of $x_1, x_0, w_0$. The transformation I am interested ...
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1answer
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random variable probability problem

I am trying to find the answer to a mathematical probability problem. let a box contain $5$ balls : $2$ balls white, $2$ balls green, and $1$ red ball (we can't differentiate between the balls by ...
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36 views

Given two random variables and two ranges, what is the probability of them being within a specified range from each other?

Given two random variables $X$ and $Y$ where $X \in [a, b]$ and $Y \in [c, d]$, $a < c < b < d$, what is the probability of $X$ and $Y$ being withing $Z$ units from each other? For example: ...
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The relationship between random variables, distribution functions and probability measures

Given a probability space $(\Omega,\mathcal{F},P)$, and a random variable $X\colon\Omega\to\Bbb{R}$, we can associate with it its distribution function $F\colon \Bbb{R}\to[0,1]$ defined as \begin{...