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

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2answers
25 views

What does $\sim$ in $X\sim \mathcal{N}(\mu,\sigma^{2})$ really mean?

This is a bit of a silly question, but I can't seem to find the answer anywhere. I feel like $X\sim \mathcal{N}(\mu,\sigma^{2})$ means that $\sim$ is a relation, but if it is a relation, what ...
1
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1answer
11 views

Finding a formula for a probability density function

The 75th percentile of a random variable X is the value X=k such that 75% of the observed values of X are less than k. For example, if the 75th percentile on an exam is 87, then 75% of the scores are ...
3
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0answers
32 views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
2
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2answers
21 views

how to understand the generation of cauchy distribution from uniform distribution?

I am learning some basic idea on generating cauchy distribution from uniform random generator $u \in [0, 1]$. I know it was discussed before in How to generate a Cauchy random variable, but during my ...
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0answers
18 views

Characteristic Function limit to 0

When calculating the limit of the following characteristic function $$\left( \frac{1}{n+1} \right)\left[ \frac{1-exp\left( \left(n+2 \right)it \right)}{\left( 1-exp(it) \right)} \right]$$ and ...
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0answers
5 views

Closed form expression for the S-transform of a random variable?

I'm trying to compute the S-transform as described in this review article on random matrix theory (section 2.2.6). They define it as $\Sigma_X(\gamma) = -\frac{\gamma+1}{\gamma}\eta_X^{-1}(1+\gamma)$ ...
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1answer
28 views

Proving the variance of pareto random variable equals (a*lambda)/((a-1)^2*(a-2))

So my PDF for the Pareto distribution is: $$\dfrac{a\lambda^a}{x^{a+1}},\quad x\ge\lambda$$ To find the variance, you need to find the integral of $x^2\dfrac{a\lambda^a}{x^{a+1}}$ and subtract it from ...
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2answers
22 views

Solution to a covariance problem

Roll two dice, let $X$ be their sum, and $Y$ be the second roll subtracted from the first. Compute $\text{Cov}[X,Y]$. Brute-force calculating $\text{Cov}[X,Y]=E[XY]-E[X]E[Y]$ is fairly horrible, ...
0
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1answer
9 views

Proof that 2 geometric random variable is NB

can someone write me the proof of 2 geometric variable are negative binomial ? $X\sim G(p)$ and $Y\sim G(p)$ how can i proof that $Z=X+Y \sim NB(2,p)$?
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1answer
50 views

Conditional expectation, $X = \varphi (Y)$

Show that if $$\forall \omega \in A \ : \ X(\omega) = \varphi(Y(\omega)), \ \ A \in \Sigma_Y$$ (that is, the equality is true for $\omega \in A$), then $$\mathbb{E}(X|Y)(\omega) = \varphi(Y(\omega)) ...
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2answers
32 views

How to calculate the probability distribution function (PDF)?

Sorry for the dumb question, I've been struggling with understanding the probability distribution function formula, what does "x" and "d" stand for in the formula , and how to use the formula? I've ...
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0answers
7 views

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) ...
0
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0answers
22 views

Conditional expectation random variable

Could you explain to me how to find $\mathbb{E}(X|Y) (\omega)$ defined as the only (up to sets of measure zero) random variable satisfying $$\Sigma_Y \subset \Sigma_{\mathbb{E}(X|Y)}$$ $$\forall B ...
6
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1answer
28 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 \ ...
1
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3answers
37 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 ...
0
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1answer
30 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
54 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) := ...
0
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2answers
26 views

Can someone help with this elementary example? [Absolutely Continuous Random Variables]

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 ...
0
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0answers
15 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 ...
1
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1answer
22 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
23 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 ...
2
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2answers
42 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
18 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
41 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 $V = C^1 \cap L^2$ ...
0
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1answer
33 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
40 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 ...
0
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3answers
22 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
22 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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0answers
25 views

Find $dE(x)$ where $dx$ is defined [on hold]

I'm confused with this example? Could someone explain this? $dx$ is defined as following: $dx = -kxdt + σdw$ why $dE(x) = -kE(x)dt + σ E(dw)$ ?
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0answers
29 views

convergence of a sequence of iid random variable [on hold]

I am given a sequence $\{ c^k X_k\}_{k\ge0}$ where $c>0$, $X_k$ are iid random variables with finite variance. For what values of $c$ does this sequence converges almost surely?
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2answers
42 views

Ito Differential Equation example [on hold]

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
36 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) * ...
1
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1answer
29 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 ...
0
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1answer
14 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
14 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
44 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
24 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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0answers
19 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 ...
0
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1answer
46 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 ...
1
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1answer
28 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. ...
0
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1answer
46 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 $P$: ...
3
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0answers
26 views

Conditional expectation, sigma algebra

Let $X$ be a random variable on $\Omega$ and $Y$ a discrete variable having values $y_1, y_2,...$. We define another random variable via conditional expectation $\mathbb{E}(X|Y)(\omega) = ...
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2answers
17 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
42 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 ...
1
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1answer
34 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 ...
1
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1answer
54 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 ...
0
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0answers
11 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 $U_{(1)}=\min U_i$. ...
0
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0answers
25 views

Random variables, indicator variables, conditional probability

Let $\{X_n \}$ be a sequence of independent identiacally distributed random variables. Let $A, B \in \mathcal{B}(\mathbb{R})$ be such that $P(X_1 \in B) \neq 0$. Let $S_n:= 1_{\{X_1 \in B \}} + ... + ...
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0answers
25 views

Binary system, number of $1$s, almost sure convergence

Could you check if my solution is correct? For $x \in [0,1]$ let $S_n$ be the number of times $1$ occurs in the first $n$ digits of $x$'s binary representation. Show that $\lim _{n \to \infty} ...
0
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2answers
33 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 + ...