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

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2answers
26 views

Uniform Distribution Problem on $X, Y, Z$

Problem: Let $X \sim \text{Uniform}(0,1)$. Let $0 < a < b < 1$. Let $$ Y = \begin{cases} 1 & 0 < X < b \\ 0 & \text{otherwise} \end{cases} $$ ...
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0answers
21 views

How to compute integrals using any probability law with Monte Carlo?

I am intrested in providing an estimation of : $\iint C(x,y)dP_X(x)dP_Y(y)$ I am able to generate random numbers from the distribution of $P_Y$ and $P_X$. Therefore I generate a big number (n=10 000)...
3
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2answers
52 views

Determine whether a random binary sequence was generated by human or natural process

Given a binary sequence, how can I calculate the quality of the randomness? Following the discovery that Humans cannot consciously generate random numbers sequences, I came across an interesting ...
3
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1answer
29 views

$\limsup$ sequence independent $\mathcal{N}(0,\sigma^2)$

In my lecture notes there is the following application of Borel-Cantelli's 2nd lemma: Let $(X_n)_{n\geq 1}$ be a sequence of independent $\mathcal{N}(0,\sigma^2)$-distributed random variables, with $\...
3
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1answer
30 views

Conditional distributions allowed pdf to take on single value?

My question is about Conditional probability distributions. From what I have learned, PDF's aren't allowed to take on singular values, yet I find that this definition seems to go out the window when ...
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1answer
27 views

Linearly Dependent Random Variables

Intuitively, what is meant to be captured by the notion of linearly dependent (real-valued) random-variables?
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20 views

What is the Standard Way to Define a Norm on a Vector Space of Real-Valued Random Variables?

Let $V$ be the vector space of real-valued random variables over $\mathbb{R}$. How does one traditionally define the norm on $V$?
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1answer
14 views

Intuitive definition of scaling random variables by a constant?

From how I understand scaling discrete random variables, we are multiplying all members in the set by the scaling constant. I.E if our random variable X = {1,2,3,4} and our scaling factor is $\alpha ...
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1answer
21 views

Scaling Random variables by a constant

First, let me state the problem: "Customers at Fred's cafe win a 100 dollar prize if their cash receipts show a star on each of the five consecutive days Monday...Friday in any one week. The cash ...
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0answers
71 views

Does this sequence of random variables converge almost surely?

I was trying to understand why almost sure convergence doesn't imply convergence of the mean and I encountered this answer. However, I do not understand why this sequence of random variables ...
1
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0answers
18 views

Weak Law of Large Numbers, biased expectation?

I want to show that: $$\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$ is a consistent estimator of $\sigma^2$. I was using the Weak Law of Large Numbers in the sense that: $$E(X_i-\bar{X })...
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2answers
65 views

Average shortest distance between a circle and a random point lying in it

What is the average shortest distance between the circle $(x-a)^2+(y-b)^2=r^2$ and a random point lying in it? This question is just idle curiosity. Basically, it's the same as finding the ...
1
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1answer
37 views

Convolution of PDFs is a PDF

Suppose $f$ and $g$ are PDFs of real-valued random variables. Show that the convolution $f\ast g$ of $f$ and $g$ (defined below) is also a PDF. $$(f\ast g)(x)=\int_{-\infty}^\infty f(y)g(x-y)\,dy.$$ ...
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1answer
46 views

How to simulate a delta-correlated random process

I'm trying to do the simulation described in the paper attached, but there is something I don't understand. The author says that the random variables which satisfy the relation (Eq. (4) in the paper) ...
0
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1answer
38 views

Probability Density Function of Random Variable which is Sum of other Random Variables

Let $X_0, X_1, X_2, ..., X_n$ each be non-identical independent random variables. Let $x_0, x_1, ... , x_n$ be possible values of each of those random variables. Let $\newcommand{\Pdf}{\...
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1answer
32 views

Probability Density Function of Random Variable which is Max of other Random Variables

Let $X_0, X_1, X_2, ..., X_n$ each be non-identical independent random variables. Let $x_0, x_1, ... , x_n$ be possible values of each of those random variables. Let $Pdf_{x0}(x_{0}), Pdf_{x1}(x_{...
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0answers
20 views

Showing independence of two independent identically distributed random variables.

The title might be confusing, here is the task: $X$ and $Y$ are two independent and identically distributed random variables with $\mathbb{P}(X=-1)=\mathbb{P}(X=1)=0.5$ and $Z=X\cdot Y$. Show that $...
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0answers
30 views

Given the joint pdf of $X$ and $Y$, find the joint pdf of $W = X+Y$ and $T = X-3Y$.

Let the two-dimensional random variable $(X, Y)$ be whose joint probability function is: $f(x,y) = 1/4$, for $0\le x\le2$ and $0\leq y\leq 2$ a) Calculate the joint probability density ...
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1answer
31 views

Calculating PDF from Autocorrelation

I have a statement like this; A zero mean Gaussian random process $X(t)$ is wide sense stationary with the auto-correlation function $R_x(\tau) = 4e^{-2|\tau|}$ And I want to find the ...
1
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1answer
19 views

Tight sequence of rv's such that $V(X_n) \rightarrow +\infty$.

Let $(X_n)$ be a tight sequence of real valued rv's, i.e. $\displaystyle \lim_K\sup_n P\left(\left|X_n\right|>K\right)=0$, defined on a common probability space, such that $E\left(X_n^2\right)<+\...
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1answer
37 views

Probability of failure of individual components in parallel and sequential system

Not sure If I'm calculating these probabilities correctly was wondering if someone could lead me in the right direction otherwise. Probability of a channel working properly is 0.8 and all channels are ...
3
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1answer
66 views

Are there general methods that can be applied when using the Borel-Cantelli Lemma, to get a statement about a sequence of random variables?

I hope the title in itself is clear, if not allow me to give an example. In Class my Professor did the following: Given a sequence $(X_n)_{n \in \mathbb{N}}$ of non-negative i.i.d. RV $X_n \sim X$...
2
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0answers
66 views

Coin-toss game with \$1 entry fee and \$3 payout [closed]

Imagine a coin-tossing bet game. You pay \$1 to play the game (for one toss), and if you win you get a prize of \$3. The \$1 to play is not refunded. The probability of winning and losing is equal. If ...
2
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3answers
66 views

Finding the density for $\min\{X, Y\}$

Problem: Let $X$ and $Y$ be independent and suppose that each has a $\text{Uniform}(0,1)$ distribution. Let $Z = \min\{X, Y\}$. Find the density $f_Z(z)$ for $Z$. Hint: It might be easier to first ...
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0answers
16 views

Using of Ito formula

I have a task- $\alpha,\beta \in R$ and $N(t)=e^{\beta t}\cos(\alpha W(t)).$ It is necessary to calculate $E[\cos(\alpha Z)]$, where Z is a standard normal random variable. I know that it is ...
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1answer
25 views

Distribution table for number of white balls without replacement

I have a box with $4$ white and $2$ black balls. I want to find the random variable distribution table for the number of white balls when I take $3$ balls without replacement. Now if we had ...
2
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0answers
161 views

Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
1
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1answer
34 views

Source of faulty reasoning in expectation of product of random variables?

For iid $\xi_i>0$, with $\mathbb E[\xi_i]=1$, what is $\mathbb E[\prod_i^M\xi_i]$? Approach 1: $\mathbb E[\prod_i^M\xi_i]=\prod_i^M\mathbb E[\xi_i]=1$. There is another approach for $M\gg1$ with ...
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0answers
19 views

Generating Failure rate function using Octave.

I'm not familiarized with programming using mathematics. My problem is described below. The lifetime T of a device has pdf Find the failure rate function. I have learned that Failure rate ...
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0answers
25 views

Conditional probability and its expectation of continuous random variables

I have two subtle questions on the definition of conditional probability and its expectation. Here's the thing. Question 1. Definition of a conditional probability density function $f_{X|Y}(x,y)=\...
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1answer
19 views

Most likely value of negative binomial random variable

If $X$ is a negative binomial random variable let's say with $p =0.2$ and $r = 4$ then how can we calculate most likely value of $X$? I thought it is expected value but that is $20$ and I guess most ...
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0answers
17 views

Does $ P(X=x|W=\tilde{w}, P(Y=y|W=w)=a) $ make sense?

I have the following question. Consider a discrete random variable $X:\Omega\rightarrow \mathcal{X}\subset \mathbb{R}$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Consider other two ...
0
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1answer
18 views

Maximal distance between random variables in interval

Suppose there are two random variables $A, B$ constrained in the intervals $A \in [a_{min}, a_{max}]$ and $B \in [b_{min}, b_{max}]$. $a_{min}, b_{min} \leq 0$ and $a_{max}, b_{max} \geq 0$. I ...
0
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1answer
48 views

Proof of linearity for expectation given random variables are dependent

The proof of linearity for expectation given random variables are independent is intuitive. What is the proof given there they are dependent? Formally, $$ E(X+Y)=E(X)+E(Y)$$ where $X$ and $Y$ are ...
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1answer
25 views

Finding the distribution of a random vector in a conditional probability problem [closed]

Players A and B are playing a game of drawing coins from two boxes without returning/replacing them. Box1 has three coins with values 0, 1 and 2 and Box2 has two coins with values 1 and 2. In the game,...
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0answers
34 views

existence of some variance over chebyshev's inequality

From the basic knowledge I have, there must exist the variance for some epsilon greater than zero but less than 1 for which the Chebyshev's inequality holds. Now the scénario I have to verify using ...
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1answer
33 views

Proof of Chebyshev's inequality for a geometric random variable

I have learnt the Chebyshev's inequality for a continuous case like log-normal and normal distributions and in trying to understand the application I came across the question: For a geometric ...
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1answer
23 views

What is the closest apporoximation for pdf of log-normal distribution?

I am unable to compute a complex integral which uses the pdf of log-normal distribution. Hence, I want to replace the pdf of log-normal distribution with an alternate function(s) (piece-wise ...
1
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2answers
33 views

What is the distribution of the subtract of two random variables?

Definition) A stochastic process $\{X(t), t \geqslant 0\}$ is said to be Brownian motion process with drift coefficient $\mu$ and variance parameter $\sigma^2$, if it satisfies that $X(0)=0$. $\{X(t)...
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0answers
44 views

Three random variables with exponential distributions

Having $X$, $Y$ and $Z$ as three independent identical random variables all having exponential distribution $E(X)=E(Y)=E(Z)=\frac{1}{\lambda}$, What is the answer of the following probability: $P(X+Y&...
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1answer
32 views

Can continuous random variables ever have positive probability on a single point?

From a textbook: Continuous random variables can lead to confusion. First, note that if $X$ is continuous then $\mathbb{P}(X = x) = 0$ for every $x$. But then later: Let $F$ be the CDF for a ...
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2answers
34 views

Are these definitions of a continuous random variable equivalent?

In a textbook I'm reading: A random variable $X$ is continuous if there exists a function $f_X$ such that $f_X(x) \ge 0$ for all $x$, $\int_\infty^\infty f_X(x) dx = 1$ and for every $a \le b$, ...
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4answers
41 views

Convolution: Give a proof that $f_T(t)=\int_{-\infty}^{\infty}f_X(x)f_Y(t-x)dx$ where $f_T(t)$ is the PDF of random variable T

Here is the question: Let $X$ and $Y$ be independent, continuous r.v.s with PDFs $f_X$ and $f_Y$ respectively, and let $T=X+Y$. Find the join PDF of $T$ and $X$, and use this to give a proof that $...
1
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1answer
37 views

Change of Uniform Continuous Variable

Let $X$ be a $U(-1, 1)$ random variable, we define $Y = X^4$. Calculate the correlation coefficient between both variables. Are they uncorrelated? PS. I don't know how to use MatJax equations, I'm ...
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0answers
17 views

Expected values of Cereal box - Linearity of expectation puzzle [duplicate]

A toy is randomly put in a given Cereal box as a promotional gift. There can be N different types of toys and each one can be of any type N (IID). (a) Find the expected number of cereal box one has to ...
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0answers
16 views

Do characteristic functions characterize the independence of random variables? [Solved] [duplicate]

It is well known that the probability density function characterizes the independence of random variables in the following sense. $$X,Y \quad\text{independent}\iff f(x,y)=f_x(x)f_y(y)$$ where $f$ is ...
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0answers
8 views

Why does the mean centered autocorrelation have a slope of -1?

I'm fundamentally not understanding something about the autocorrelation function (as defined by numpy.correlate). Let's say I create a bunch of random signals $s_1, ...
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2answers
70 views

Interpretation of correlation (coefficient)

In an discussion we were confronted with a very special opinion about correlation in respect of financial assets. The widely used correlation coefficient is used here to give an idea about how ...
2
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1answer
64 views

What is the pdf of sum of log-normal and normal distribution?

The question goes like this: $Z = X+Y$; where $X$ is Log-normal Random variable with parameters - $\mu = 0 \quad \sigma^2= 1$, $Y$ is Gaussian Random variable with $\mu= 0\quad \sigma^2= 1$ What is ...
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0answers
28 views

Notation: should Markov chains steps be noted by uppercase or lowercase letters?

I'm reading the chapter about perfect sampling of the "Monte Carlo Statistical Methods" by Robert and Casella, 2004. I've got an issue about notation, when they talk about random mappings, they say $$...