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

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how can find statistical and linear independence from sample space

Let sample space is $ S=\{\varsigma_1,\varsigma_2,\varsigma_3,\varsigma_4\} =\{-1, -0.5,0.5,1\}$ and define two random variables as $ X(\varsigma)=1/\varsigma $ and $Y(\varsigma)=2^{-\varsigma}$. a) ...
6
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1answer
45 views

Divergent series of random variables

I've been trying to prove that given a sequence of independent random variables with identical distribution $\{X_n\}_{n \in \mathbb{N}}$ such that $P(X_1 \neq 0)>0$, so also $P(X_i \neq 0) >0 \ ...
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3answers
58 views

How to explain why the probability of a continuous random variable at a specific value is 0?

Consider X as a continuous random variable which can assume any value in [0, 1]. It is known that P(X=x)=0 where P is the probability density function. I want to understand this intuitively. The math ...
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1answer
45 views

2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
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4answers
239 views

Expected value of the minimum of a non-negative random variable and a constant

X is a non-negative random variable. Define Y = MIN(X, c) where c is a constant. What is E[Y]? I am modeling the constant as another random variable whose pdf is Dirac Delta function: $f_{c}(x) := ...
2
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1answer
71 views

Sequence of finite, positive and i.i.d random variables and limit of $\frac{S_{n+1}}{S_{n}}$

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of finite, positive and i.i.d random variables and let's call $\mu:=E(X_{1})>0$ and $S_{n}:=\sum_{i=1}^{n}X_{i}$. We know that ...
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2answers
52 views

CDF of $-\ln X$ where $X$ is uniform on $(0,1)$

I'm having difficulty studying this part of the subject, because i can't get through this first example, can anyone help? Let $$X: U(0,1)$$ Find the distribution function of the following random ...
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0answers
20 views

In a group of 2n couples , in a random fashion 2n people are chosen.

In a group of 2n couples , in a random fashion 2n people are chosen. Find the distribution of the random variable X , which represents the number of couples in the chosen group. $\Omega$-probability ...
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1answer
37 views

Let $X$ and $Y$ be two uniformly distributed random variables on $[0,1]$. Find $E(X^k)$ and $E(XY^k)$.

Let $X$ and $Y$ be two uniformly distributed random variables on $[0,1]$. Find $E(X^k)$ and $E(XY^k)$. How can you do this? Do you need the change of variables technique? I am a bit confused about ...
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1answer
28 views

continuous probability density functions

Continuous distributions assign probability 0 to individual values. But, according to DeGroot, it doesn't mean that it is impossible for the random variable to take individual values. So, why not make ...
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2answers
106 views

Expectation of quotient of random variables

Let $X_1,...X_n$ be independent, identically and non negative random variables, and let $k\le n$. Compute: $$E\left[{\sum_{i=1}^k X_i\over \sum_{i=1}^n X_i}\right]$$ this question has already been ...
2
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1answer
30 views

Random number generator from a piecewise PDF

I'm trying to create a random number generator on the interval $(a,c)$ given a probability density function defined as: $$f(x) = \left\{ \begin{array}{lr} \dfrac{C}{x} &, x \in (a,b)\\ ...
2
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2answers
56 views

Is there a meaningful way to approximate a discrete random variable?

Is there a meaningful way to find a continuos approximation of a discrete random variable? Thoughts for the $L^2$ case If $X \in L^2$, then we may want to consider the subspace $V = C^1 \cap L^2$ ...
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1answer
36 views

Would it be safe to say that a random variable X is identially zero when its first and second moments are both zero?

Would it be safe to say that a random variable $X$ is identially zero when its first and second moments are both zero? If it is true, how would you prove this? This step is needed when we prove that ...
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1answer
53 views

How do we approximate sum of random variables?

Suppose we have independent, identically distributed random variables $X_n \notin L^1$. I would like to approximate, in some way, the distribution of their sum $\sum X_n$ .The problem is that these ...
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3answers
28 views

Equality in distribution

If $A\stackrel{d}{=}C$ and $B\stackrel{d}{=}D$, is it $A+B\stackrel{d}{=}C+D$, where $A,B,C,D$ are dependent random variables?
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2answers
29 views

Determining a mean and skewness of a probability density function

Is there any way to determine the mean and skewness of a probability density function WITHOUT integrating? I have the following function: $$ f(x)= \begin{cases} x&\text{if}\ 0\le x\le1,\\ ...
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2answers
57 views

Ito Differential Equation example [closed]

Could someone explain Ito through an example as following? How to use Ito differential equation to find $dy$ , where $y = e^{w(t)}$
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0answers
39 views

Exponent - Solving for an unknown within an expectation

I have reached a stage where I need to solve for an unknown number, $\theta$ . However, I stuck and don't know how to proceed further. The equation to be solved is: $E\left[ \exp(\theta a^i) * ...
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1answer
45 views

Calculating the distribution of a compound random variable

Given $X\sim U(1, 0)$ and $Y\sim Exp(1)$, determine the density function of $Z:=\frac{X}{Y}$. Now, without looking up how to do it I tried to figure it out myself. The value of the density function ...
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1answer
23 views

Variance of not quite the product of two independent, normally distributed random variables

Let's say I have two independent variables, $X\sim N(10,9)$ and $Y\sim N(5,4)$. $X$ represents the number of orders received in a month, and $Y$ represents the size of each order. For this example, a ...
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1answer
19 views

Probability function and distribution - taking out fish from a pool

In a pool of fish there are 4 fish of type A, 3 fish of type B, 2 fish of type C, 1 fish of type D. We take out fish without returning them until we get fish of type C for the first time. ...
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0answers
65 views

Game of Keno from Sheldon Ross Chapter 4

I am facing with the following problem: A game popular in Nevada gambling casinos is Keno, which is played as follows: Twenty numbers are selected at random by the casino from the set of numbers 1 ...
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0answers
43 views

What's the difference between a random variable and a measurable function?

I've tried to wrap my head around the measure theoretical definition of a random variable for a couple of days now. In his book Probability and Stochastics, Erhan Çinlar defines a measurable function ...
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27 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...
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1answer
82 views

Mean return time in Markov chain

Given the following Markov chain: $p_{0,1}=1$ (if we are in state 0, we must go to state 1) $p_{i,i+1}=p_{i,i-1}=0.5$ There are infinitely (countably) many states. I assume that $X_0=0$ and define ...
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1answer
32 views

Random variable and distribution - number of tests a teacher has to make

$100$ students do a test. The probability of failing the test is $0.6$, those that failed, do a retest, the probability of failing the retest is $0.5$. Those that fail the retest do another retest. ...
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1answer
91 views

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: ...
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2answers
25 views

Random Variable being $F$-measurable

It is said the Random variable is $F$-measurable if $\{X\leq x\}$ is an element of $F$. Is $X$ not $F$measurable once it is not less than or equal to $1$ $x$ or only for all?
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2answers
96 views

Inverse of a mean, exponential distribution, expected value

Could you help me find the expected value of this random variable? Let $X_1, X_2, ... $ be independent identically exponentially distributed with parameter $\lambda$ random variables. What is the ...
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1answer
66 views

How to generate integer random numbers that equal to another random number?

I am running a simulation in Excel, and need to generate a group of integer random numbers summing up to another random integer, how can I possibly do it? For instance I have an integer random number ...
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1answer
61 views

What is the probability that a multivariate Gaussian random variable is greater than zero?

I am looking for a way to find the probability that $p(x > 0)$, where the vector $x$ has a multivariate Gaussian distribution $$ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \sim ...
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0answers
15 views

Distribution of $\sum n_i(U_i-U_{(1)})$

Let $U_i$ be independent random variables with pdf $f_i(x)$ ($i=1,\ldots,k$) where $$f_i(x)=\frac{n_i}{\sigma}\exp(-\frac{xn_i}{\sigma}), x>0$$ Let $n=\sum n_i$ and $U_{(1)}=\min U_i$. ...
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2answers
38 views

Finding an unbiased estimator for a parameter, dicrete variable

Let $X : \Omega \to \mathbb{N}$ be a random variable. Define $p_i = P(X=i), \ \ i \in \mathbb{N}$. Find an unbiased and consistent estimator for $p_1$. I need to find an estimator $\alpha_n(X_1 + ...
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1answer
32 views

Random Variables. A Probability Question.

We have the folloing problem in my probability class, and I want to know if I have outlined it correctly. In a Kingdom there are $3$ prisoners; $A,B,$ and $C$. The king says that two are condemned to ...
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1answer
62 views

How can you find $P(\frac{X}{Y-X}<0)$ if $X\sim Geometric(p)$ and $Y\sim Bernoulli(p)$

Let the independent random variables $X\sim Geometric(p)$ and $Y\sim Bernoulli(p)$, I want to prove that $P(\dfrac{X}{Y-X}<0)=(p-1)^{2}(p+1)$. Do I need the joint probability mass function for ...
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2answers
49 views

To use or not Bernoulli trials

I was asked to model the following experiment: Consider the n-th toss of a fair coin, and the event $E$ = '$k$-th toss results in heads'. I find easier to model the experiment using n random ...
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0answers
19 views

Support for a linear combination or transformation of random variables

Let $X, Y \sim iid U(0,1)$ and $c_1, c_2 \in \mathbb{R}$. In the linear combination $Z = c_1X+c_2Y$, we know that the probability density function of $Z$ depends on the relationships of $c_1$ and ...
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1answer
17 views

“cover the unit sphere by c-fine grid” to prove the vector length preserved by random projection?

The below figure is extracted from the paper http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4031351 . I did not understand the techniques used in the proof, namely, 1."cover the unit ...
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0answers
43 views

Distribution Function Of a Random Variable X - Question

This is a homework question pretty much but I do not understand how to approach it. The distribution function of the random variable X is given: F(X) = 0, x < 0 x/2, 0 <= x < 1 2/3, 1 ...
0
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1answer
70 views

How to calculate the probability distribution function (PDF) and the cumulative distribution function (CDF)?

Sorry I'm a novice to both functions and just didn't get a clue how to solve this problem (having been reading the theories for the whole day but still ...) The problem is: We have now two investment ...
2
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0answers
23 views

Max cut problem

I've just looked at the standard proof using the probabilistic method stating that the max cut problem has a lower bound of $|E|/2$ for any graph $G=(V,E)$. More specifically if $X$ is the random ...
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1answer
60 views

Trying to understand the behaviour of i.i.d.

In a course called introduction to probability theorem we are covering now i.i.d. (independent and identically distributed random variables). I already know when two variables are independent: $X, Y$ ...
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0answers
39 views

Characteristic function of an asymmetric Laplace distributed random variable

What is the characteristic function of a random variable with density $$f_X(x) = \frac{1}{2} [ 1_{x>0} \, a e^{-a x} + 1_{x<0} \, b e^{b x} ], \; \; \; \quad a,b > 0 \quad \quad ? $$ My ...
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0answers
22 views

When is a coupling ''natural''?

The definition of coupling is written below. In some articles, I found the term "natural coupling". When is a coupling said to be ''natural''? Definition of coupling between two random variables: Let ...
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1answer
44 views

p.d.f. of a position variable from stochastic velocity p.d.f.

I have a stochastic process, $v(t)$, that represents a velocity, and has a known probability distribution function $f(x,t)$ which is time-varying. I am interested to acquire a probability ...
0
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1answer
35 views

Non-standard question about random variables

I am not sure which subbranch of mathematics this is, so I cannot give a precise tag. I am doing research, and this suddenly popped out of no where. So, please hear me out. $x$ is a variable that ...
0
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1answer
17 views

new bounds for transformed random variable

Let $Y \sim U\left ( 0,1 \right)$, I have already determined the new pdf for the transformation $Z=Y^2$. I used the cdf technique for this. So the new pdf for $Z=Y^2$ is $f_Z(z) = ...
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1answer
67 views

Probability of tail event using Kolmogorov's 0-1 law

If $X_1,X_2,... $ are independent random variables and $X=\sup_nX_n$ then $P(X<\infty)$ is either 0 or 1. I think that if we prove the event to be a tail event then the result will follow. But I ...
0
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1answer
44 views

Variance and Expected value of internet connection

I am working on a probability/statistics problem! The problem is as follows: Your internet connection is very poor. It constantly alternates between being functional for x minutes and being down for ...