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

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0
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2answers
25 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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1answer
24 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 ...
1
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2answers
34 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 ...
0
votes
0answers
24 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." ...
0
votes
2answers
128 views

Find the conditional expectation $E[X_2|F_1]$.

$X_n$ is a sequence of random variables. Let $X_0 = 4$, $X_n=2X_{n-1}$ with prob $= 3/4$ or $X_n=0.5X_{n-1}$ with prob $= 1/4$. $F_i$ is a filtration of the sigma field. $F_0 = \{$null,$\omega\}$, ...
4
votes
1answer
169 views

Can two random variables $X,Y$ be dependent and such that $E(XY)=E(X)E(Y)$?

Can someone define independence of two random variables with this "product rule", or are there any counterexamples?
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0answers
26 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 ...
1
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1answer
13 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 ...
2
votes
0answers
35 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 ...
0
votes
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 ...
1
vote
1answer
34 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 ...
2
votes
3answers
49 views

Let $X\sim\mathrm{Exp}(1)$, and $Y\sim\mathrm{Exp}(2)$ be independent random variables. Let $Z = \max(X, Y)$. calculate $E(Z)$

Here's a question I'm trying to solve: Let $X\sim\mathrm{Exp}(1)$, and $Y\sim\mathrm{Exp}(2)$ be independent random variables. Let $Z = \max(X, Y)$. calculate $E(Z)$ I'm can't understand ...
2
votes
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 ...
1
vote
1answer
713 views

Given x is an exponential random variable, find median & probability

For the median, I believe that I should integrate the function, ∫x0λe−λtdt=1−e−λx Then I need 1−e−λm=.5 for m, which is equivalent to e−λm=.5. m=ln(2)/λ =>m=ln(2)/.2
-2
votes
0answers
15 views

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

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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
2answers
53 views

How can I prove that expectation of conditional random variable?

I know the following results are true. However, I forgot to prove them. Please let me know how to prove them. $$E(X)=E(E(X|Y))\tag1$$ $$P(X)=E(P(X|Y))\tag2$$ (1) \begin{align} ...
1
vote
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 ...
0
votes
1answer
45 views

Expected value of X and Y for a given problem [closed]

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
votes
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. ...
2
votes
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 ...
2
votes
1answer
22 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 ...
2
votes
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 ...
1
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0answers
22 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 $Z_t X_u$?
0
votes
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 $\mathbf{C_{xx}}\neq\mathbf{I}$, ...
1
vote
0answers
37 views

Sequence of random variables, mean zero, convergence to -infinity

What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ...
1
vote
0answers
28 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 ...
0
votes
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: ...
0
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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 ...
1
vote
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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votes
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 ...
0
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0answers
12 views

Given an event field, is there a random variable generating it? [duplicate]

In probability space $(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, for any event field $\mathcal{G}\subset\mathcal{F}$, there always exists a random variable $X$, such that $\sigma(X)=\mathcal{G}$? Is ...
0
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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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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 ...
1
vote
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 ...
0
votes
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 ...
0
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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 ...
0
votes
2answers
54 views

CDF of sum of 3 dependent random variables

Given three dependent random variables, $S_1, S_2$ and $S_3$, such that $0 < S_1, S_2, S_3 < \infty$ and assuming known their joint PDF $f_{S_1,S_2,S_3}(s_1,s_2,s_3)$ I would like to find the ...
0
votes
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] ...
3
votes
1answer
86 views

Statistically Independent Random Variables

Problem: For the statistically independent random variables $X$ and $Y$ with $f_X(x)=1$, $1\leq x\leq 2$, and $f_Y(y)=e^{-(y-1)}$, $ 1\leq y< \infty$, determine $f_Z(z)$ where $Z=X+Y$. I ...
1
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2answers
30 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?
0
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0answers
48 views

Measure theoretic definitions of random variables/probability distributions

I'm asking your help to answer questions (a),(b),(c) outlined in the summary below. The questions are so connected that I found difficult to ask them separately. Could you also let me know if you find ...
1
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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 ...
0
votes
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 ...
2
votes
1answer
391 views

Function of stationary processes

Suppose we have stationary processes $X_1(t), X_2(t),..., X_n(t)$ and let $f_t(X_1(t), X_2(t),..., X_n(t))$ be a continuous function of these stationary processes. Will $f_t(\cdot)$ also be stationary ...
3
votes
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, ...
2
votes
1answer
24 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 ...
1
vote
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 $YZ$?