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

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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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2answers
26 views

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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4answers
36 views

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 ...
1
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1answer
28 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, ...
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0answers
24 views

Identification of the probability distribution of a discrete random variable and knowledge of its support.

I am confused on the following issue regarding the identification of the probability distribution of a discrete random variable and the knowledge of its support. Let $X$ be a random variable defined ...
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0answers
17 views

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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0answers
15 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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5 views

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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2answers
41 views

Probability density function for product and minimum of i.i.d. $U(0,1)$ random variables

If $U$ and $Y$ and $Z$ are i.i.d. $U(0,1)$ random variables, find the pdf for $A= U \times Y$ and $B = \min \{ U,Y,Z\}$.
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2answers
60 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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0answers
26 views

Given X and Y ind. rv's, when is f(X,Y), g(X,Y) ind.?

I have to parallel questions. I was trying to solve this one: "Given two independent real-valued randomvariables X and Y defined on the same sample space, is it true that X and X+Y are independent." ...
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1answer
38 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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0answers
27 views

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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1answer
14 views

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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0answers
43 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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0answers
13 views

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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0answers
17 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 ...
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0answers
15 views

Lower bound on probablity for a bounded random variable using only expectation of it. [closed]

I found this question am not able to proceed with it. Please help me with it. Let 0 < ε and δ < 1, and let Y be a random variable ranging in the interval [0,1] such that E(Y) = δ + ε. Give a ...
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2answers
33 views

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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0answers
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 ...
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2answers
40 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. ...
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1answer
41 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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0answers
23 views

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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1answer
23 views

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
12 views

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 $\mathbf{C_{xx}}\neq\mathbf{I}$, ...
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1answer
51 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 ...
0
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1answer
19 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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0answers
10 views

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
26 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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1answer
25 views

Coefficient of correlation between linearly related random variables [closed]

A random variable $X = -3Y+10$, where $Y$ is also a random variable and has zero mean. Are they correlated? What is the correlation coefficient in this case? I know that it equal ...
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0answers
29 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 ...
2
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1answer
38 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
21 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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1answer
34 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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1answer
31 views

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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4answers
124 views

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 ...
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0answers
94 views

Mathematics Homework 2 Question 8d :What is the probability you and your partner are now able to meet the new deadline? [closed]

You are working on a programming project with your partner for a computer science course. The project is due in 48 hours. Together, you are to produce a computer program and each of you are assigned ...
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1answer
21 views

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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2answers
83 views

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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2answers
33 views

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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0answers
28 views

Integral equation with a random variable [closed]

Suppose we have an integtal of this kind: $$\displaystyle Q(\tau)=\int_0^\tau dt K(w(t)+\nu(t))$$ where $\nu(t)$ is a gaussian noise $\nu(t)=\mathcal{N}(0,\sigma)$ Because $w(t)$ is a known function, ...
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1answer
34 views

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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3answers
35 views

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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1answer
25 views

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
15 views

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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0answers
33 views

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
42 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 ...
2
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1answer
41 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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3answers
70 views

$Y$ 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 $Y$. ...