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

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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
26 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 $\mathbf{C_{xx}}\neq\mathbf{I}$, $\mathbf{C_{yy}}\neq\...
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1answer
56 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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1answer
21 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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11 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
28 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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33 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
40 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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1answer
35 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
32 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
125 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 $\mathbb{...
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1answer
22 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
34 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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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
40 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
27 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
16 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
38 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
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 ...
2
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1answer
49 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
73 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 $Y$. ...
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1answer
16 views

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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2answers
45 views

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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0answers
18 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
30 views

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 ...
2
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1answer
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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36 views

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{...
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2answers
55 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} E(E(X|Y))&=\...
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0answers
13 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 ...
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1answer
15 views

Almost sure convergence of the inverse

If a sequence of non-negative random variables $X_1, X_2, \dots$ converges almost surely to a random variable $X$, that is $X_n \xrightarrow{a.s} X$ or equivalently $P(\lim\limits_{n\to\infty}X_n=X)=1$...
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0answers
40 views

Problem that may use Borel Cantelli Lemma

So, there is a sequence of identically distributed independent random variables taking values on the integers, and they have a positive expectation. The problem is to prove that with probability 1 the ...
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2answers
51 views

Probability of rectangles area being less than 0.5 w/ total length of sides = 2

Question: A random point splits the interval [0,2] in two parts. Those two parts make up a rectagle. Calculate the probability of that rectangle having an area less than 0.5. So, this is as far as I'...
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1answer
29 views

Distribution of the product of two lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas. Consider the corresponding log-normal random variables: $...
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1answer
32 views

Expected sum of the draws by Mr. B?

Ms. A selects a number X randomly from the uniform distri- bution on [0,1]. Then Mr. B repeatedly, and independently, draws numbers Y1,Y2 ,.... from the uniform distribution on [0,1], until he gets a ...
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1answer
44 views

Let $(X_i)$ be i.i.d. exponential, is the set $\{X_1,X_2,\ldots\}$ almost surely dense in $(0,\infty)$?

To be clear, I'm asking if the range of the random sequence $(X_i)$ is dense in $(0,\infty)$ a.s. I thinks the answer is yes, because for any $0<a<b<\infty$, we have $$P(X_1 \notin (a,b), ...
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45 views

Integrating a Random Variable and establishing the maximum of a related function

Frequency Regulation of a Power Grid I have a battery that is used to regulate the frequency of a power grid. That is, as the grid frequency varies about it’s ideal value, $f_{nom}$, the battery ...
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1answer
25 views

Characteristic function of discret random variable

I try to show the following: Suppose $(X_n),n\geq1$ is a sequence of random variables with uniform distribution on $\{1/n,\dots,n/n \}$. Show that $(X_n)$ converges in distribution to a random ...
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0answers
16 views

Simple and partial correlation

(i) The partial correlation coefficient $r_{12.3}=r_{12}-r_{13}r_{23}/(\sqrt{(1-r_{13}^2)(1-r_{23}^2)}$ and the simple correlation coefficient $r_{12}=\sum{(x_{i1}-\bar{x_1})(x_{i2}-\bar{x_2})/\sqrt{\...
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1answer
42 views

A probability transformation

Let $X,Y,Z$ be continuous random variables; $Z$ are independent of $X,Y$. Is the following transformation right ? \begin{align} P(X,Y \in (a,b),Y+Z \notin (a,b))&=\int_a^bP(X \in (a,b),y+Z \notin (...
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1answer
32 views

Use the delta method to find the distribution of $Z_n$

Let $\overline{X}_n=\overline{X}$ the sample mean such that $\sqrt{n}\overline{X}_n\rightarrow^D N(0,1)$ where $\rightarrow^D$ means converge in distribution. Use the delta method to find the ...
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1answer
52 views

Conditional probability when conditioning on continuous-discrete variables

I am confused on the notion of conditional probability when the conditioning variable is continuous. Consider the random variables $X,Y$ on the probability space $(\Omega, \mathcal{F}, P)$ with ...
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20 views

Arbitrary vs. random subsets: computing probabilities

Let $G=([n],E)$ be a graph having minimum degree $\delta(G) \geq (1-\delta) n$. For some $q=q(n)$, let $G_q=([n], E_q)$ be the random subgraph of $G$ obtained by deleting each edge independently with ...
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32 views

Is a “deterministic” subset of a random subset random?

Let $S$ be some set and consider $X \subseteq S$ of size $|X|=x$ u.a.r. (among all the subsets having this size). Now, use some properties of this set $X$ to find some subset $Y\subseteq X$ of some (...
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22 views

Compute probability that a random subset has a certain property (when we know probability for an arbitrary subset)

Suppose we have a ground set $[n]:=\{1, \dotsc, n\}$. Now, we pick a random subset $S \subseteq [n]$ u.a.r. among all the subsets of $[n]$ having size equal to $s$. In general, if we know that for ...
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1answer
18 views

What is the probability distribution of the following random variable?

Let $A^n$ and $B^n$ be independent random variables taking values in $\{0, 1\}^n$. Let $Y^n = A^n + B^n$ (Hence, taking values in $\{0, 1, 2\}^n$). How can we express the distribution of $Y^n$ in the ...
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1answer
41 views

How to show $P(|X-E(X)|\leq x)=1\implies V(X)\leq x^2$

Let $X$ be a random variable with finite variance. I am trying to show if $P(|X-E(X)|\leq x)=1$ then $V(X)\leq x^2$. Could somebody please help me correct my working? $|X-E(X)|\leq x\iff(X-E(X))^2\...
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1answer
35 views

Variance of a random variable [closed]

How do you get the variance of a random variable $X$ where $X = \frac{1}{6}(A \cdot B)$ and where $A$ and $B$ are two independent random variables with variances $\sigma_A^2$ and $\sigma_B^2$, ...