Questions tagged [random-variable]

A random variable or stochastic variable is a value that is subject to chance variation (i.e., randomness in a mathematical sense).

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KL divergence between Laplace distributions

Let $p(x_i) = \mathcal{L}(\mu_i, \sigma_i)$ and $q(x_i) = \mathcal{L}(\mu_i', \sigma_i')$, where $\mathcal{L}$ corresponds to the density of a Laplace distribution. Let $\mathbf{x} = \begin{bmatrix}...
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Is the realization of random variable also a random variable?

In class, a teacher told me that the realization of a random variable is also a random variable. For example, if I take the a sample mean, and that mean results in the value 35, then 35 is also a ...
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Finding probability involving dependent random variables [closed]

Suppose train on line A arrives in time uniformly distributed between 0 and 4mins, train on line B arrives in time uniformly distributed between 0 and 6 mins, and furthermore the time interval between ...
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Ratio of cubic and quadratic form in random variables is approximately normal?

Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which is a ratio of a cubic form and a ...
rgvalenciaalbornoz's user avatar
2 votes
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Distribution IID uniform variables given their ranking [duplicate]

Description Let $N\in\mathbb{N}^{+}$ and $X_{n}\stackrel{IID}{\sim}U(0,1)$ for $n\in\{1,...,N\}$. Given $X_{1}\leq X_{2}\leq X_{3}\leq...\leq X_{N}$, I would like to understand $f_{X_{n}}$ by writing ...
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Series representation of the Incomplete Beta Function [closed]

What is the series representation of the incomplete beta function, for example, if $X$ is a non negative random variable having the following CDF \begin{align} F_X(x)=I\left(x; a,b\right); \end{align} ...
learning statistics 's user avatar
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Parameter estimation of a WSS process

As my research revolves around parameter estimation from signals that evolve in time in a random fashion, I am curious to know what features/ retrievals people normally use to determine the parameters ...
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Calculating $E[(\sum X_i)^4]$

Trying to figure where I'm going wrong with the following. My goal is to calculate var$(\bar X_n^2)$ using $E[(\bar X_n)^4]=\frac{1}{n^4}E[(\sum X_i)^4]$ given that $X_1,...X_n$ are iid with $EX_1=\mu,...
reyna's user avatar
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If $X \sim\textrm{ Bin}(100, 0.5), $ then what is the approximate distribution of $(X/5 - 10)^2? $

I am not able to solve this using transformation since binomial does not have a CDF. The question has 4 options, so I tried calculating the expectation of this and then comparing it to the ...
Anweshan Goswami's user avatar
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How does a Random Sample relate to Random Processes and Random Variables?

What is the difference between a Random Sample, Random Variable (RV) and Random Process (RP)? As far as I know, a RV is a mapping from an experimental space to the real numbers and a RP is a mapping ...
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Seeking Lower Bound for Partition Probability in Random Variable Analysis

I am reaching out to seek assistance with a probability problem involving random variables. For each $p$ in $[1,\infty)$, consider positive random variables $X_{1,p}, X_{2,p}, \ldots, X_{n,p}$ such ...
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Reference for Moments of Gamma Distribution random variable [duplicate]

I want a reference that explain the $n^{th}$ moment of the gamma random variable having shape and scale parameters for the gamma distribution, specifically the following moment equation \begin{...
learning statistics 's user avatar
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Prove that the equality holds [closed]

How to prove that for any random variables $X$, $Y$ and $Z$ with finite variances, we have $Cov(X,Y)=E(Cov(X,Y|Z))+Cov(E(X|Z),E(Y|Z))$?
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Poisson distribution in two disconnected region

Suppose $N$ is a RV such that$$N\sim Pois(\lambda)$$ and $N_{1}$ and $N_{2}$ are two Random Variable distributed in two disconnected region called $R_{1}$ and $R_{2}$ respectively and $N=N_{1}+N_{2}$ ...
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What is the distribution of a RV with the constant random variable? [duplicate]

For random variables (rv) $X$ and $Y$ on a space $\Omega$: Assume the rv $X\sim f_0$ distributed and $Y(t)=c$ is a constant rv, i.e. $Y\sim \delta(t-c)$ using the $\delta$-distribution as a short ...
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Taylor approximation for function of a random variable [closed]

There is a function $f$ whose domain is the space of CDFs on $\mathbb{R}_+$ and whose range is $[0,1]$, e.g. $f$ maps a CDF on to a real number. Further, $f$ is continuous, increasing with respect to ...
user's user avatar
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If $A^2$, and $B^2$ are DEPENDENT random variables, will $A$, and $B$ be necessarily DEPENDENT too?

I know that if $A$, and $B$ are independent, the independence is preserved for $A^c$, and $B^c$, where $c$ is a constant. I am wondering if the same applies to the case where the random variables are ...
Roberto's user avatar
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Test to see if expected value of a random variable is infinite

Say I have a finite real valued random variable, $X(\theta)$ where $\theta$ is some set of parameters such that the distribution of $X$ depends only on $\theta$. In particular, depending on $\theta$, ...
roundsquare's user avatar
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Obtaining confidence intervals from composite defined functions

Assume you have a set of $n$ independent random variables $X_1, X_2, \dots, X_n$ with unknown distribution and mean (finite) values $\mu_1, \mu_2, \dots, \mu_n \in \mathbb{R}$. Moreover, there are $n$ ...
gabriel2029's user avatar
1 vote
1 answer
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Choosing the right statistical analysis for my data

I have 20 field plots. In each plot, I have taken between 1000–1500 measurements (continuous values) of a given variable using an instrumental device and recorded information qualitatively through ...
Darius's user avatar
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The training error of best hypothesis

Let $\mathcal{X}$ and $\mathcal{Y}$ denote the domain set and label set respectively. Also let $\mathcal{D}$ be a distribution over $\mathcal{X}$ and $f:\mathcal{X} \to \mathcal{Y}$ be the true ...
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The Issue of High Correlations and Maximal Random Structure in Mixed Effects Regression

I have two questions and I'd really appreciate your advice on them. Firstly, I'm looking into the impact of sugar intake on the weight of both children and adults. However, there's an issue because ...
Elizabeth's user avatar
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CLT, Euclidean Distances and Means

This may sound like a rather odd question but I have stumbled upon a distribution of means that resembles a normal, but I am not sure whether it is possible to prove that it is indeed one or not. Here ...
DarkenExcalibur's user avatar
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Joint density of two functions of a uniformly distributed random variable

I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$. I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
johnsmith's user avatar
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Stochastic order symbols - Intuition [closed]

I apologize for the following set of questions, which may seem trivial. Let $X_n$ denote a sequence of random variables. Then, $X_n = o_p(1)$ means that $\lim_{{n \to \infty}} X_n = 0$. I can use ...
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3 answers
321 views

$\chi^2$-distribution of $(x_1-x_2+x_3+x_4)^2-(-x_1+x_2+x_3+x_4)^2$ with $x_i \sim N(\mu_i,\sigma_i)$

Given are 4 independent normal variables $x_i \sim \mathcal{N}(\mu_i,\sigma_i)_{i=1,\ldots,4}$, that are used to define the random variable $$Z\sim (x_1 - x_2 + x_3 + x_4)^2 - (-x_1 + x_2 + x_3 + x_4)^...
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Moments and PDF of solution to random quadratic equation

Consider the following random quadratic equation, $$ x^2 + Z x + Y = 0, $$ where, $$ \begin{gathered} Z \sim \mathcal{N}(\mu_Z,\sigma_Z), \qquad Y \sim \mathcal{N}(\mu_Y,\sigma_Y). \end{gathered} $$ ...
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An example of a random variable $y\in L^\dagger_2$ having more than one linear combination, $y = \Sigma_{i}\alpha_i x_i = \Sigma_{i}\beta_i x_i$

In the answer for the following exercise: Let $\{x_1,...,x_n\}$ be a finite collection of random variables with $E(x_i^2) \lt \infty$ ($i = 1,..., n$). Show that the set of all linear combinations $\...
Tran Khanh's user avatar
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Expected steps until collision of a simultaneous walk in two random functional graphs

Let $f$ be a function $f : \{0, 1, 2, \dots, n - 1\} \to \{0, 1, 2, \dots, n - 1\}$. Now let $u \in \{0, 1, 2, \dots, n - 1\}$. $u$ is our initial value. Consider the sequence $$ u, f(u), f(f(u)), \...
Finn R's user avatar
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2 votes
1 answer
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When are Random Variables easier to use than Sample Spaces?

Background This is related to the following question: Is an experiment's Sample Space always useful? I understand that Sample Spaces are sets holding all the possible outcomes of an experiment. ...
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Why are Random Variables used instead of Sample Spaces? [duplicate]

Background This is related to the following question: Is an experiment's Sample Space always useful? I have been reading about Sample Spaces, and I understand they represent all possible outcomes ...
Connor's user avatar
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Order of centroids across two independent random variables

Say I have a function $q_{D, \Theta}: \mathcal{X} \to \mathcal{Y}$ that depends on independent random variables $D$ and $\Theta$. I want to consider "centroids" of $q$ with respect to ...
ngmir's user avatar
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3 votes
1 answer
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Expectation of product of sample averages

I have a bunch of iid random variables $X_i\sim q$ and I have defined other random variables $A_i = a(X_i)$ and $B_i = b(X_i)$. Then I bumped into the following expression $$ \begin{align} \mathbb{E}\...
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4 votes
1 answer
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Is the relation "not FOSD" transitive?

$X:\Omega\to [0,1]$ is a random variable. It is known that first order stochastic dominance FOSD is a partial order that is transitive: $X$ FOSD $Y$, $Y$ FOSD $Z$ implies $X$ FOSD $Z$. Now consider ...
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1 vote
1 answer
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Variance in set of different random variables

Imagine we have sample of values a_1, a_2,...a_n, with a_i value originating from a normal distribution with mean mu_i_a and variance var_i_a, thus each value is single realization of different random ...
GAMer's user avatar
  • 123
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0 answers
23 views

Misunderstanding on the use of Popoviciu and von Szokefalvi Nagy's inequalities on the variance of a unbiased estimator

Let $X_1,\cdots,X_n$ be (discrete in my case) i.i.d. and bounded between $m$ and $M$. I'm interested in bounding the variance of an unbiased estimator: $$\mathbb{V}\left[\frac1n\sum_{i=1}^nX_i\right]$$...
Tristan Nemoz's user avatar
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"This parameter is redundant" - SPSS Binary Logistic mixed model

I know there is a question that asked about a similar warning, but that user's "error" had to do with dummy coding while I think my problem has to be with statistical power and understanding ...
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Resample random variable to fit different variance

Suppose I have samples drawn from a random variable, and I want to multiply that random variable with a scalar constant. How should I transform the samples such that they would have been drawn from ...
mroelofs's user avatar
1 vote
1 answer
93 views

Expectation of the reciprocal of a standard normal random variable [duplicate]

If $\mathbf{X} \sim_{iid} \mathcal{N}(\mu, 1)$ then we know that the sample mean $\bar{X} \sim \mathcal{N}(\mu, 1/n)$, how would we show that $$\mathbf{E}\left(\frac{1}{\bar{X}}\right) = \infty $$ and ...
delta_99's user avatar
1 vote
0 answers
31 views

Problem with the expectation of transformed random variable [duplicate]

I read a paper where the random variable $z$ is assumed to follow a log-normal distribution. \begin{equation} \mathbb{E}\left(z^{1-\chi}\right)=\int_{0}^{\infty}z^{1-\chi}\frac{1}{z\sqrt{2\pi s^{2}}}e^...
optimal control's user avatar
2 votes
0 answers
125 views

Does a sequence of Bernoulli random variables with parameter $1/n$ converge to $0$ almost surely?

Consider independent random variables $X_n\sim\operatorname{\mathsf{Ber}}\left(\frac1n\right)$. The problem is whether $$X_n\xrightarrow{\mathrm{a.s.}}0$$ is true or not. I tried to use the definition ...
ultralegend5385's user avatar
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34 views

Why isn't X treated as a random variable in linear regression MLE? [duplicate]

I am very confused by this because when I watch videos or read about MLE with linear regression it seems to be commonly assumed that $X$ is fixed or that if it is random we don't care for the purposes ...
AdmiralMunson's user avatar
1 vote
1 answer
139 views

Show that for random variable $X$ with $N = \{1, 2, \ldots \}$, $E(X) = \sum_{n = 1}^\infty P(X \geq n)$ [duplicate]

Prove that for random variable with natural numbers from 1 to infinity the expected value $E(X)$ is equal to $\sum_{n = 1}^\infty P(X \geq n)$. Is this the mathematically correct way to prove it? And ...
Ste0l's user avatar
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0 answers
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If random variables X,Y are independent is $P(X>k)*P(Y>z)=P(X>k,Y>z)$? [duplicate]

If random variables X,Y are independent is $P(X>k)*P(Y>z)=P(X>k,Y>z)$? I know if X and Y are independent then $P(X=k)*P(Y=z)=P(X=k,Y=z)$
Coo's user avatar
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1 vote
0 answers
93 views

If $X_1, \dots, X_n$ iid, are $f(X_1), \dots, f(X_n)$, also iid? [duplicate]

If I have independent and identically distributed random variables $X_1, \dots, X_n$, then are $f(X_1), \dots, f(X_n)$ themselves independent and identically distributed? I think the answer is yes, ...
caitlin's user avatar
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1 vote
0 answers
26 views

Random Variable with E and V

Question is Let X be a random variable with $E(X)=1$ , $V(X)=5$. I need to find (a) $E[(1+X)^2]$ and (b) $V(4+3X)$. I think I solved (b) alright. Please let me know if there is an error. For (b), I ...
DLo9's user avatar
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0 votes
0 answers
22 views

If A and B are correlated, what is the correlation between |A1 + A2| and |B1 + B2|?

To preface, let random variables $X = A_1 + A_2$ and $Y = B_1 + B_2$. $A_1$ and $A_2$ are copies of $A \sim N(0, 1)$, and $B_1$ and $B_2$ are copies of $B \sim N(0, 1)$. In this situation, if $A$ and $...
Vance's user avatar
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0 answers
51 views

Lower bounding weighted sample variance

Let us assume that we draw a sample $\{X_i\}_{i=1}^N$ from a random variable $X$ and we have a discrete probability distribution $q_{ij}$, i.e. $0 \leq q_{ij}\leq 1$ and $\sum_{ij} q_{ij} =1$ (the $...
raskolnikov's user avatar
4 votes
1 answer
138 views

Minimum Pearson's correlation between $X$ and sign($X$)$\cdot X^2$

Suppose $X$ is a (whatever bounded or not) continuous random variable ($X$ is not constant) with an arbitrary distribution. Is it possible to construct a distribution s.t. $\text{Corr}(X, \text{sign}(...
cat's user avatar
  • 53
2 votes
1 answer
88 views

For a normal distributed random variable X what is the distribution of c/X

Assume $X\sim \mathcal{N}(\mu, \sigma^2)$ For a normal distributed random variable $X,$ what is the distribution of $c/X$? I had a look at ratio distributions but could not find it. PS: The issue ...
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