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

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42 views

Can someone explain this question about random variables?

In each of the following questions, a random variable is described. Identify the random variable as either binomial, Poisson, geometric, negative binomial or hypergeometric and describe a reasonable ...
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1answer
32 views

Coupling of r.v.

I am trying to answer this question. If $X$ and $Y$ are random variables on $(\Omega, \mathcal{B})$, show \begin{align*} \sup_{A \in \mathcal{B}} |P[X \in A] -P[Y \in A]| \le P[X \neq Y]. ...
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30 views

Sum of $\{X_n\}$ iid random variables contained in a compact interval implies each $X_i=0$ a.s.?

I am working through an exercise that starts with a sequence i.i.d. random variables where for $a\leq b$, $$\Pr\left(\lim\sup_n \sum_{i=1}^{n} X_i \in [a,b] \right) \neq 0.$$ Does this require $X_i ...
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0answers
18 views

Constraint Optimization

I have a sequence of iids defined by: $f(x|\theta) = \exp(-(x-\theta))\;\;\;\; \theta<x<\infty$ To find the maximum likelihood estimate, i should maximize the log likelihood with respect to ...
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1answer
12 views

Probability in continuous functions. ( Simple Question )

Completed the question but can't get my head around part e? Can someone help?
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19 views

Conditional independent random variable in graphical models vs measure theory

When we say random random variables $X$ and $Y$ are conditionally independent given and $Z$, I understand it to mean given the $\sigma$ algebras generated by $X$,$Y$ and $Z$ denoted ...
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1answer
34 views

How can I prove that Xn converges to 0 in distribution?

Xn~U[-1/n,1/n]. Since for convergence in distribution Xn-->X iff Fn(x)-->F(x). First of all, I am trying to get the cumulative and it is Fn(x)=(1+xn)/2. Hence, am I doing something wrong?
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2answers
88 views

Independent Exponential Random Variables

I am currently trying to figure out a problem and it is using notation that I have never seen before so I am pretty stuck, any suggestions would be greatly appreciated! Let $X, Y, Z$ be independent ...
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20 views

$P(X\le z)$ for $z$ between $z\in(1,\infty)$ for binomial distribution

If I have a binomial distribution of $X$ (a random variable), where $X=\{I:X_1= \dots X_{i-1}=0, X_i = 1\}$, how do I find an expression $P(X\le z)$ for $z\in(1,\infty)$? Any help appreciated!
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1answer
18 views

Upper bound of min separation between n randomly chosen points in a figure

Given a rectangle of size $6\times12$, prove that if $7$ points in it are chosen uniformly at random, the distance between at least $2$ points is $\leq 5$. I don't know how to approach this problem. ...
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1answer
23 views

Random sample of size $n = 2k$, calculate $p(X_1 < 1/2, X_2 > 1/2, X_3 < 1/2, X_4 > 1/2, \dots, X_{2k} > 1/2)$.

A random sample of size n = 2k is taken from a uniform pdf defined over the unit interval. Calculate $p(X_1 < 1/2, X_2 > 1/2, X_3 < 1/2, X_4 > 1/2,...., X_{2k} > 1/2)$. Solution: Then ...
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1answer
58 views

Good ways to sample $n$ identical and dependent random variables

I'm wondering if there's a good way to talk about sampling identical but dependent random variables where it's also easy to see how the distribution evolves as we move from $n$ random variables to ...
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1answer
37 views

For a real-valued random variable it holds: $ E(|X|)<\infty\Leftrightarrow \sum_{n\in\mathbb{N}}P(|X|>n)<\infty$

Let $(\Omega,\mathcal{A},P)$ be a measurable space and $X$ be a real-valued random variable on $\Omega$. I want to show that it holds: $$E(|X|)<\infty\Leftrightarrow ...
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0answers
8 views

How to find the variance of N which is a combination of two random variables?

Given that X~Poisson(m=4) we define Y as the possiblity to succeed after X tries. so Y~B(X,0.5). N=2Y+X E[X]=4 , E[Y]=2 E[N]=2*E[Y]+E[X]=8 Now we need to find the variance of N. I tried ...
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0answers
28 views

Computing a Finite Expectation

Assume $1\leq\ k<m<n$ are positive integers and $X_1,X_2,...X_n$ are i.i.d. Geometric($p$) random variables. For all $j\geq\ k$ define $I_j=[(i_1,i_2,...,i_k):1\leq\ ...
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1answer
33 views

Differential of two geometric brownian motions

I am currently taking a finance course which includes some math that is currently above my level, it is however not a pure math class and we are just supposed to be able to apply the math to the given ...
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0answers
13 views

Why the doubly non-central F distribution does not have a mean or variance if the denominator degree of freedom is less than or equal 2 ??

Normally the doubly non-central F distribution is generated by the division of two non-central chi squared Random Variables,, so what is the the problem of using any famous formula to get the mean of ...
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1answer
34 views

A hand of six cards is dealt from a standard poker deck. Find formula for p_(XYZ) (x,y,z).

A hand of six cards is dealt from a standard poker deck. Let X denote the number of aces, Y the number of kings, and Z the number of queens. a) write a formula for p_(XYZ) (x,y,z). b) Find ...
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2answers
58 views

Frankie and Johnny game. What should Johnny strategy if he wants to minimize his expected loss?

Frankie and Johnny play the following game. Frankie selects a number at random from the interval $[a, b]$. Johnny, not knowing Frankie’s number, is to pick a second number from that same inverval and ...
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2answers
71 views

A random variable $X$ uniformly distributed over the interval $[0, 2\pi]$

A random variable $X$ distributed over the interval $[0, 2\pi]$ a) the pdf of $X$ b) the cdf of $X$ c) $P(\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ d) $P(-\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ ...
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0answers
25 views

Gaussian random vector with 0 mean [duplicate]

Let $X =(X_1,X_2,X_3,X_4)$ be a Gaussian Random Vector with $\mathsf E(X_1)=\mathsf E(X_2)=\mathsf E(X_3)=\mathsf E(X_4)=0$. Show that $$ \mathsf E(X_1 X_2 X_3 X_4) = \mathsf ...
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1answer
42 views

Poison distribution variance,probability. and mean.

Let $X$ be the poisson random variable such that $P(X = 2) = 9P(X=4) + 90P(X=6)$ a) find the mean and variance of $X$. b) find P(X $\geq 1$) c) find P(X $\leq 10$) Ok so for the first question I ...
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1answer
13 views

How is this Variance found in this old question?

On this question: Probability: Normal Distribution they find these values: $\hat\mu = .05(150) = 7.5\space,\hat\sigma = \sqrt{150(.05)(.95)} = 2.67$ I see how they got $\mu$, but how did they get ...
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2answers
55 views

Motivation behind study of martingales

Today I wanted to ask a question which I am sure has been answered in multiple places but for which I do not yet have a very clear understanding. Though martingales is a very well explored area of ...
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1answer
36 views

find the mean and variance of this poisson random variable

Let $X$ be the poisson random variable such that $P(X = 2) = 9P(X=4) + 90P(X=6)$ find the mean and variance of $X$. I'm not sure how to approach this problem..am i supposed to multiply each ...
2
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1answer
12 views

Find mass function with 3 dice and 3 different Xs

There are $3$ dice you roll one at a time, $X$ is the number of distinct numbers, as in, $X=1$, you have $(1, 1, 1)$ since there is $1$ distinct # $X=2$, $(1, 2, 1)$ or $(2, 1 ,1)$ etc... $X=3$ all ...
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1answer
37 views

Finding the probability of a randomly selected event?

I know I'm over-thinking the following question, I just need to know how to start! In a certain population of women 4% develop symptoms of a classic disease, 20% are smokers, and 3% are smokers and ...
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1answer
27 views

Represent probability with multiple distributions. Archer shooting bullseyes problem.

The goal is to come up with two ways to represent this probability: An archer shoots a bulls-eye with probability $0.4$. If the archer shoots ten arrows, what's the probability that at least 3 are ...
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1answer
24 views

Limit of a jointly independent sequence of random variables

Can I say the following? If a jointly independent sequence of random variables $X_1,X_2,\dots$ converges to random variable $X$ in the mean square sense, then $X$ is independent of the elements of ...
2
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2answers
25 views

Expected value of X-x for exponential distribution

Assume $X\sim$ exponential$(\lambda)$. In class we noted that $E[X-x|X\geq x]=\frac{1}{\lambda}$. Why is this? I would have thought that $E[X]-E[x]=\frac{1}{\lambda}-x$.
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1answer
21 views

Random Distance on Torus

Let $U=(X_U, Y_U)$ and $V=(X_V, Y_V)$ be two independent random points in $[0,1] \times [0,1]$, where each possible position is equally likely. Now I am interested in the probability that these two ...
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2answers
28 views

Find $E[\max (R_1, R_2)]$ when $R_1$ and $R_2$ are independent and uniformly distributed in $[-1,1]$

Find $E[\max (R_1, R_2)]$ when $R_1$ and $R_2$ are independent and uniformly distributed in $[-1,1]$. So first I was thinking something along the lines of $$P(R_1 = n, R_2 \leq R_1)$$ would be ...
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1answer
59 views

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]?

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]? $E[e^{\frac{2X}{3}} - 3] = \int_0^2 \! e^{\frac{2X}{3}} - 3 \, \mathrm{d}x$ $= \frac{3}{2}(e^{\frac{4}{3}} - 5) = -1.8095$ I am integrating over the ...
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1answer
22 views

How to express combined discrete-continuous RVs in one pdf?

Let's say we have a random variable $X$ that behaves in two different ways where $X\sim$Bernoulli(1/3) AND $X\sim U(0,1)$. $X$ follows the Bernoulli distribution 25% of the time and the uniform ...
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1answer
16 views

How do you say that variable is randomly chosen with a random distribution for range [3, 42]?

This question is only about how to formulate something in English for a bachelor's thesis in computer science. I have a variable $x$ which is randomly initialized. It is chosen from a (continuous) ...
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27 views

Stochastic processes

Update I am a bit confused whether $y_t$ is independent over time under the following assumptions: Consider, first a RV $A$, that follows this process: $A_t = \rho A_{t-1} + e_t$, where $e_t$ is ...
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22 views

How do I simplify $\sum_i^n(y_i-rx_i)^2$, where $r = \frac{\sum y_i}{\sum x_i}$?

I want to simplify: $$\sum_i^n(y_i-rx_i)^2$$ where $y_i$ and $x_i$ are random variables and $r = \frac{\sum y_i}{\sum x_i}$. I've tried expanding the summand and replacing $r$ with $\frac{\sum ...
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0answers
13 views

Conditional Expectation of a vector of random variables given another vector of random variables

Let $\{X_i\},\;i=\{1,2,3,4,5\}$ be a set of 5 continuous random variables . Then how do I calculate $\mathbb{E}[(X_1,X_2)\mid (X_3,X_4,X_5)]$ where $(X_1,X_2),(X_3,X_4,X_5)$ are two vectors of given ...
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1answer
39 views

Conditional expectation almost sure

If $X_1 = X_2$ on a measurable set $B \in \mathfrak F$ then $E(X_1\mid\mathfrak F)=E(X_2\mid\mathfrak F)$ almost sure on $B$.
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11 views

maximization of a function with random variable

I would like to know whether this is true in general, and if not when this can be. I am not sure and so I am mostly asking for confirmation. So, is the following correct ? $$\log [\max_{x} ...
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2answers
77 views

Uniform distribution, as a sum of biased Bernoulli trials.

Suppose that the probability of $x=0$ is $p$, and the probability of $x=1$ is $1-p=q$. Consider the random sequence $X=\{X_i\}_{i=1}^{\infty}$. We map this sequence by $C$ to a point in the interval ...
2
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1answer
40 views

If the variance is $0$ is it constant?

We know that the variance of a constant is $0$. Is the converse also true? Can we say that if the variance of some random variable is $0$ it is a constant?
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1answer
37 views

Conditional expectation constant on part of partition

I have a question about conditional expectation, while looking for the answer here on stackexchange I noticed that there are a few different definitions used, so I will first give the definitions I ...
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1answer
73 views

Probability function of X and Y when two balls are drawn with no replacement

Two balls are drawn at random from a box containing ten balls numbered 0, 1, ... , 9. Let random variable X be the larger of the numbers on the two balls and random variable Y be their total. ...
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1answer
44 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
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0answers
42 views

Relation between minimum and sum of two random variable

I am interested in finding a relation that involves two independant random variables, that could be used to describe the sum of these, or the minimum of these. For example, regarding the sum, we know ...
5
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0answers
54 views

Applying PCA on covariance matrix in order to generate a new random variable.

Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. ...
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1answer
25 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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1answer
31 views

Expectation of a powered complex circular gaussian process

Assuming a complex circular zero-mean gaussian random process (or vector) $\textbf{x}$ $\left(\textbf{x}\sim \mathcal{CN}\left(0,\sigma^2\right)\right)$. $\mathbb{E}\{\textbf{x}\}=0$. The question ...
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0answers
49 views

Probability that one random variable is greater than or equal to another

Assume $X$ and $Y$ are i.i.d. with exponential distribution with parameter $\lambda = 1$ (the probability density functions $p_X (x) = e^{-x}$ and $p_Y (y) = e^{-x}$ in $[0, +\infty)$, $0$ otherwise). ...