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

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20 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
18 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
35 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
22 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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0answers
22 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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0answers
39 views

Computing conditional expectation variable given variables X, Y

When it comes to conditional expectation, I can compute $\mathbb{E}(X|Y)$ when I know the distribution of $X, Y$ when they are continuous or discrete. But I don't know how to find $\mathbb{E}(X|Y)$ ...
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31 views

Expected value of conditional expectation, discrete variable

We are given a random variable $X$ on $\Omega_1$ and a discrete variable $Y: \Omega_2 \to \mathbb{N}$. We consider $\mathbb{E}(X|Y)$ as a random variable defined as follows: $$\mathbb{E}(X|Y)(\omega)= ...
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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
39 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
33 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
45 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
10 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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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} ...
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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 + ...
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0answers
19 views

Variance of sample variance

We are given $\{X_i \} $ iid random variables with $\mathbb{E}X_i = \mu$ and $D^2X_i < \infty$. I'm trying to compute $D^2(\sigma^2_n)$ where $$\sigma^2_n= \frac{1}{n} \sum_{i=1}^n (X_i - \mu)^2$$ ...
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0answers
18 views

Expectation of inverse of sum of random variables, exponential distribution

I have a question similar to this one: Expectation of inverse of sum of random variables only my variables have exponential distribution. So $X_1, X_2, ...$ are $iid$ with exponential distribution. ...
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1answer
40 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
48 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
17 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
13 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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25 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 ...
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1answer
36 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 ...
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0answers
15 views

Martingale difference sequence [closed]

Show that the sequence of random variables $w_t = (u^2_t - \sigma^2)x^2_{t-1}$ is a martingale difference sequence.
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0answers
21 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
52 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
14 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
17 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
33 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 ...
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0answers
19 views

Adding x+y gamma function [closed]

I know how to set up a independent variable when it's normal. And I'm sure they are somehow related but I'm not sure where to start on this problem. It's not covered at all in the book I have and he ...
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1answer
34 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 ...
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1answer
13 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
45 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 ...
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1answer
41 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 ...
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1answer
49 views

How long would it take to a lottery number repeat?

In Professor Stewart’s Cabinet of Mathematical Curiosities the following is asked: You have $1000$ songs on your MP3 player. If it plays songs ‘at random’, how long would you expect to wait ...
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3answers
50 views

Probability of success in $n$ trials

I'm stuck on my statistics homework and would appreciate your help. Question: Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is $0.12$. ...
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0answers
17 views

Distribution of a r.v. with the same mean and variance is abs. cont. with resp. to the normal distr.

I have a question concerning the Kullback-Leibler divergence or relative entropy. In a book I found the following definition of the KL-divergence: Let $(\Omega, \mathcal F)$ be a measurable space. ...
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2answers
30 views

Joint Random Variable: Given f(x,y), find P(X>Y)

There are 2 continuous random variables, X and Y. Say the joint pdf of (X,Y) is f(x,y). How do you find the P(X>Y) generally? Like I am not sure where to start with.
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1answer
13 views

diagonalizing a matrix with random elements

Consider the matrix $A = \begin{pmatrix} cY & 0 \\ 2 & 1\end{pmatrix}$, where $c \in \mathbb{R}$ and $Y$ is a random variable that is uniformly distributed over $[0,1]$ (That is, $Y \sim ...
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1answer
31 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let ...
2
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1answer
52 views

Show that a Markov Chain is ergodic

Let $Y_n$ be iid random variables with values 1,2,3..n so that $P[Y_i=j]=p_j>0$, where $i\leq1$ and $1\leq j\leq n$. I think I managed to show that $Y_n$ is a Markov chain using the definition, ...
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0answers
27 views

Applying chain rule in probability?

Let $X,Y$ be random variables with distribution functions $F_X(x)$, $F_Y(y)$. Let $W(u,v)=max\{0,u+v-1\}$. why can we take the following limits "inside" $W$? $lim_{(x,y)\to ...
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0answers
22 views

Does normalization of a random vector, destroy uniformity?

If I have a random vector in Rn that has a uniform distribution in the domain [a,b]n, a<0, b>0. Is uniformity lost or preserved (in the unit sphere) if I normalize the vector (using the euclidean ...
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0answers
25 views

Prove that the variance of a discrete random variable increases with a parameter

I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$. Let ...
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0answers
14 views

perfect coin is tossed n times. Let Sn denotes the number of heads obtained. What is the expectation of Sn?

The Problem is: A perfect coin is tossed n times. Let Sn denotes the number of heads obtained. What is the expectation of Sn? I got to E(Sn) = $\sum_{n=1}^{+\infty} \space\space\space Sn ...
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0answers
19 views

Convergence of vectors

Recently I've read a paper and there is one moment I cannot fully realise on my own. It states as follows. There is a vector of estimates $\hat{\mathbf{X}} = (\hat{X}_1, \dots, \hat{X}_N)$ (N is ...
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1answer
32 views

The solution to this joint distribution problem is too terse for me to understand.

I was wondering if I could get clarification on the following problem: We know that $\sum_x\sum_y f(x, y) = 1$. Then $4\theta_1 + 6\theta_2 = 1$. I understand that $P[X = 1] = ... = P[Y = 4] = ...
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1answer
38 views

Is c parameter or constant (random variable X with given density)

problem: is c constant or parameter solution for this is to $ \int_{1}^{2} cx^2 dx = \frac{7c}{3} $ $ \int_{2}^{3} cx dx = \frac{5c}{2} $ Until now I understand what is going on; next (I am ...
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1answer
33 views

Calculating inter-arrival times and arrival times of a Poisson process

For a practice exam in stochastic processes I have to answer the following questions. Let $\{N(t): t\geq 0\}$ be a poisson process with rate $\lambda$. Let $T_n$ denote the n-th inter-arrival time ...