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

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1answer
790 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
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0answers
11 views

Variance of a complex random variable

The variance of a complex random variable $ X$ with average $\langle X\rangle =0$ is defined as $$\sigma_X^2 = \operatorname{Var}[\vec X] = \langle X^* X \rangle$$ Here $^*$ is complex conjugation. ...
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3answers
34 views

Independence and expected value

I have a theorem that says If two random variables $X,Y$ are independent, then for any non-negative measurable functions $f:E\to\mathbb{R}$ and $g:E\to\mathbb{R}$ the following holds ...
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2answers
49 views

Find the mathematical expectation [on hold]

Find the expectation of $$f(x) = a(1+x)^{-(1+a)}, \quad x>0.$$ The answer given is $\frac{1}{a-1}$. I am not getting the answer. Please help.
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2answers
111 views

Can someone explain what a portfolio is in financial math?

I took mathematical probability last semester and now I am taking financial mathematics, but only probability was a pre requisite for financial math (no finance classes were required). These types of ...
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1answer
33 views

How do can i solve the integral, finding cdf [on hold]

Let $X$ be an exponential random variable with mean 1 and Y a uniform random variable between $0$ and $1$. Assume X and Y are independent and let $Z =e^{X/2}$ Find the joint cumulative ...
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1answer
32 views

Sum of random variables goes to infinity

I'm trying to show the following: Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with $\mathbb{E}[|X_1|]<\infty$ and $\mathbb{E}[X_1]=\mu$. Consider ...
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0answers
17 views

Distribution function of Sum of IID Exponentiation Variables of Variable amount

So I'm trying to determine the distribution function of a random variable, S, give: $N \sim Geo(\frac{1}{1+\lambda}) $ $S_i \sim Exp(\mu), \forall i\in [0,N]$ $S = \Sigma^{N}_{i=0}S_i$ $S = ...
2
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0answers
33 views

Let $X$ and $Y$ be iid real-valued random variables. Show $P[|X-Y| \le 2] \le 3P[|X-Y| \le 1]$. [duplicate]

Found this question in The Probabilistic Method and tried for hours to prove it, but I'm not getting anywhere. Can anyone walk me through it? I see that if we can show $P[1 \le X - Y \le 2] \le P[|X ...
1
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3answers
82 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, ...
3
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1answer
24 views

Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
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0answers
31 views

Uniform probability bound involving two binomial random variables

Fix $c>1$. Does there exist number $m$ and function $f(\epsilon)$ such that for every $0<\epsilon<1$, $0<p<\epsilon$, and $n > f(\epsilon)$, we get ...
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0answers
25 views

When convergence a.s. implies convergence in mean?

Can someone help me with proving the following: Assume that $X_n$ converges almost surely to $X$, where $X_n$ is a sequence of non-negative random variables. Furthermore, assume that the sequence ...
2
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1answer
33 views

Exchangeability and independence of random variables

I have a question on the relation between exchangeability and independence between random variables. Consider the random vectors $$u_1:= \begin{pmatrix} \epsilon_{1}\\ \epsilon_2\\ \epsilon_3 ...
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0answers
18 views

Linear independent in random variable and observations

I am confused with some fundamental concepts. Here for $n$ random variable $X_1,\cdots,X_n$, i.i.d and follow standard normal distribution, the probability that there exists a set of constant ...
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0answers
4 views

Deriving spectral norm or similar quantity for structured random matrix

I have a problem where I have no idea to start. Suppose a simple Least Squares system with $M$ unknowns $c$ and $N$ observations $y$ which is given through the linear mapping $X$: $$y = X c$$ It is ...
1
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1answer
47 views

Questions on probability law

I'm trying to prove/disprove the following true or false statements, and I want to know if they are correct For every measurable function $g:\mathbb{R}\to \mathbb{R}$, $\mathbb{E}[g(X)]$ is ...
0
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1answer
28 views

How to handle the noise covariance matrices in a basic Kalman Filter setup?

I've recently been trying to learn about Kalman Filters; most explanations of the Kalman Filter confuse me in what is known / unknown. I'll assume the following setup: \begin{equation} \begin{split} ...
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1answer
26 views

How do you find the probability mass function of XYZ, XY+XZ+YZ, and X^2 + YZ? [on hold]

Any guidance and explanations on the following question would be greatly appreciated. The answers are provided but I am trying to understand how to go to that point. Suppose that X , Y , and Z are ...
0
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2answers
19 views

Conditional Expectation and Variance Question

So I have a question I'm absolutely stumped with: Given two random variables, $X$ and $Y$ , with common variance, $\sigma ^2$, where $\mathbb{E}(Y|X) = X + 1$, find $\rho (X,Y)$. So I obviously need ...
2
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2answers
23 views

Simple Probability - Enumeration and Geometric Distributions

I am not sure as to why this particular practice problem does not use a geometric distribution. A prize is randomly placed in one of ten boxes, numbered from 1 to 10. You search for the prize asking ...
2
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2answers
38 views

uniqueness of joint probability mass function given the marginals and the covariance

Let X and Y be two nonnegative, integer-valued random variables. Is there a way to find the joint probability mass function, i.e. $$ \mathbb{P}(X= k, Y= h) $$ for some $k,h\geq 0$, given the ...
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0answers
34 views

Question about order statistics [on hold]

$X_i, i = 1,2,3,\ldots,n$ be IID continuous RV's, with common CDF $F$. For $y \in \mathbb{R}$ define: $N(y) =$ number of $X_i$'s less than or equal to $y$ ($N(y)$ takes values $0,1,2,\ldots$ Show ...
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1answer
22 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
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0answers
17 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
0
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1answer
26 views

How to find the joint probability mass function of X and Y of a die toss?

Suppose that the die is tossed. Let X equal 1 if the result is an even number, and let it be 0 otherwise. Also, let Y equal 1 if the result is a number greater than three and let it be 0 otherwise. ...
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0answers
40 views

Is the notation $\mathbb{P}(X \cap Y) = \mathbb{P}(X,Y)$ common?

For two random variables $X,Y$, is the notation $\mathbb{P}(X \cap Y) = \mathbb{P}(X,Y)$ common? In a probability class last year we had always used $\mathbb{P}(X \cap Y)$. This year in a stochastic ...
4
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1answer
54 views
+50

On the linear combination of $\pm 1$ random variables

Let $X_1,\dots, X_n$ be i.i.d symmetric $\pm 1$ random variables, i.e. $X_j$ takes values in $\{-1,1\}$ with $$\mathbb{P}(X_j=1)=\mathbb{P}(X_j=-1)=\frac{1}{2}.$$ Let $a_1,\dots,a_n\in\mathbb{Z}$, ...
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1answer
45 views

Deriving mass/density functions of variables $\log(X)$, $X+Y$, $\operatorname{sgn}(X-1/2)$.

Could you help me with the following question? Suppose that a point with co-ordinates $(X,Y)$ is chosen uniformly from the square $\{(x,y)\in \mathbb{R}^2: 0 \leq x,y \leq 1\}$. For each of the ...
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1answer
19 views

Given Conditional Expectation check if it satisfies the density function. [on hold]

I have this math problem that I can't seem to solve. Let $X$ be a continuous random variable that only takes non-negative value that satisfies $\mathbb{E}(X\mid X \ge t) = t + ...
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0answers
45 views

how find an event with probability $6.6p^2q$ [closed]

How to find events with the following probabilities $6.6p^2q$,$6.7p^2q$,$6.5p^2q$,$3.4pq$ using repeated bernoulli trials with success probability p[$\backepsilon$ $0<p<1$]?
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1answer
28 views

Is sharing the same support a necessary condition for exchangeability?

I am confused on the meaning of exchangeable random variables. The question is: consider the random variables $X_1,X_2,X_3$ defined one the same probability space $(\omega, \mathcal{F}, P)$; is ...
1
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1answer
80 views

Conditional distribution of random variable X given itself

I'm stuck with something that might seem trivial but gives me headache. What is the distribution of $X|X$, i.e. the conditional distribution of $X$ given $X$? I'm pretty confident that: $$\mathbb ...
1
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1answer
28 views

What is the approach to understand this algorithm?

Given $\{x_1, x_2,\ldots x_n\}$ where $x_i \in \{0, 1\}$ there is a binary equation $\varphi$ that is $x_{t_1}+x_{t_2}+\cdots+ x_{t_m}=0 \mod 2$ where $t_i \in \{1,2,\ldots,n\}$ for $x≥1$, ...
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2answers
37 views

Doubt in Conditional Probability

I'm studying Information theory from the book Information Theory, Coding and Cryptography-Rajan Bose. I got confused at one pos where they have derived the equation ...
11
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1answer
300 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...
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0answers
24 views

Expected Value of $P(|Y_n^{(K)}| > \epsilon)$ where $Y_n^{(K)}$ is the random sum of a sequence of RV converging to 0 in Probability

I have been struggling with this for countless hours, I would appreciate a hint to get me going in the right direction (no complete answer please) Problem: Assume that for all $k \in \mathbb{N}$ ...
1
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1answer
11 views

Probability of intersection of multiple 2-way universal events

If given a set of events that are known to be 2-way universal, is there a closed-form solution for the probability of their intersection? If so, how would you go about finding it? I know that for 2 ...
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0answers
16 views

Coupling Brownian Motions

I want to simulate three freight rate indices which are naturally correlated. The freight rate dynamics ($X$) can be modeled as a geometric Brownian motion: $dX_{t} = \mu X_{t}dt + \sigma ...
1
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1answer
29 views

Proving that $|P\{X=m\}-P\{Y=m\}| \leq P\{X\neq Y\}$

Let $X$, $Y$ be random variables on the same probability space. Show that for all $m$, $$|P\{X=m\}-P\{Y=m\}| \leq P\{X\neq Y\}$$ I'm actually not even sure how to start. I think it's going to ...
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0answers
6 views

representative sample

population is students taking a Chemistry class; Sample of 60 students as a random sample from this population. Select one variable on the survey and argue whether or not you think this sample is ...
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1answer
34 views

Finding PDF of function of a random variable

Suppose $X$ has PDF: $f_X (x)= \lambda e^{-\lambda(x+2)}$ , for $x \ge-2$ $f_X(x)=0$ , for $x <-2$ Determine the PDF of $Y = X^2$. I am stuck because for $-2\le X \le 2$, $0\le Y \le 4$, and I ...
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1answer
11 views

Difference between random number and random variant?

After generating random number we can get the random variant by using inverse transform or other techniques. What is the difference between random number and random variant. Can anyone explain it with ...
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0answers
16 views

Multivariate normal distribution problem

Consider three Gaussian variables $X_1,X_2,X_3$ with $\mathbb{E}[X_i]=0$ and $\mathbb{E}[X_iX_j]=\rho_{ij}$ for $i,j=1,2,3$. Then, three new variables are defined: $$ \left\{ \begin{array}{l1} ...
3
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0answers
48 views

$P(|X_1+X_2|<x)\le P(|X_1|<x)$ for every independent centered continuous $X_1$ and $X_2$?

Let $X_1$ and $X_2$ be zero mean independent continuous random variables. Then, is it true that $P(|X_1+X_2|<x)\le P(|X_1|<x)$. The intuition is that summing independent variables increase ...
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1answer
32 views

What is $E[Z|Z\ge 0]$ where $Z$ is a continuous random variable with support in $[-1,1]$?

I have a random variable $Z$,I seek an expression for $E[Z|Z \geq 0]$. I assume this is easy to get hold of but I just can't seem to get it. As a further complication $Z=X-Y$, where $X$ and $Y$ are ...
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113 views

Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
2
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1answer
81 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
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3answers
53 views

Uncorrelating random variables.

I was reading this answer, and the first sentence seemed more intuitive at first than after thinking through it: If $\pmatrix{X\\ Y}$ is bivariate normal with mean $\pmatrix{0\\0}$ and ...
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votes
2answers
30 views