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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7 views

Condition for zero or non-zero bias under transformation of random variable?

Suppose $\epsilon$ is a zero-mean univariate random variable noise with finite variance. I would like to find the condition on the function $f$ so that: $$ \exists x \in R \ \mbox{s.t:} \mathbb{E}[f(...
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42 views

Product of a Gaussian by a Beta random variable

I'm trying to find the distribution of a random variable $Z = X \cdot Y$, where $X \sim N(\mu,\sigma^2)$ and $Y \sim \text{Beta}(\alpha,\beta)$ with $\alpha$=1. I have tried with the convolution ...
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Linear mixed effect model: Small and unbalanced number of repeated measures per individual

I have a data set which contains 2 or 3 repeated measurements taken from 50 individuals over a 6 year period. The time between measurements is inconsistent. My goal is to estimate the effect age (...
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Population distributions/data generating functions in Bayesian Statistics

In many frequentist stats courses, random variables come from some distribution at the population level and as such we could say that $y=X \beta + \epsilon$ is the true function for something like ...
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How can $X$ be a discrete random variable? [duplicate]

Suppose that the cumulative distribution function of discrete random variable $X$ is given by, $$F(x) = \begin{cases} 0 & \text{$x$ < 0 } \\[1.5ex] \dfrac{x}{4} & \text{$0 \leq x<1$}\\[...
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Mann-Whitney Normal Approximation process help

Let $X_{1}, X_{2}, ..., X_{n}$ is i.i.d sample from $X$ and $Y_{1}, Y_{2}, ..., Y_{m}$ is i.i.d sample from $Y$. And both samples are independent each other. Trying Mann-Whitney U-test then, $U =$ $\...
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Derivation of Pearson's Product Moment Correlation Coefficient's (PPMCC) distribution from bivariate normal variables?

I'm interested in reading through the derivation of the probability density function that PPMCC follows when it's input is a bivariate normal variables. Mathworld gives the following equalities: $$P(r)...
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2answers
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Is this problem calculable only due to the parameter choices?

I am looking at a problem form Hogg, Tannis & Zimmerman (Ed. 10), and I am curious if the given problem is calculable (for an upper-level undergrad math/stats course) because of the choice of the ...
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1answer
48 views

Show $X_n \to 0$ in probability

I am asked to show : Let $X$ be a real-valued random variable on $(\Omega, F , P)$ and define $X_n(\omega) = nX(\omega)$ if $n<X(\omega)\le n+1$ and $0$ if else. Prove that $X_n \to 0$ in ...
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Monte Carlo simulation for grouped averages [duplicate]

Assume we have $N$ random variables $X_1, \ldots, X_N$. As an example, assume that these random variables describe test scores of $N$ students. I am interested in finding the distribution of average ...
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What does $X_1,…,X_n $ mean in $X_1,…,X_n \sim N_p(0,\Sigma)$ (iid)?

What exactly does $X_1,...,X_n$ mean in $X_1,...,X_n \sim N_p(0,\Sigma)$ (iid) ? I am confused, since what I imagine is that the variables $X_1,...,X_n$ are the columns of a dataset? But From the fact ...
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If X and Y ~ Uniform(0,1), what is the distribution X/Y? [duplicate]

Given that $X$ and $Y$ are random variables drawn from a uniform distribution between $0$ to $1$, i.e. $$ X \sim Uniform(0, 1) \\ Y \sim Uniform(0, 1)$$ And given that $Y \neq 0$, what do we know ...
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44 views

Show That $\sum_{K=1}^{n}\frac{X_k}{n^{\frac{1}{\alpha }}}$ If ${X_n}$ is $X_k$s Same Distribution

Let ${\{X_n}\}$ be a sequence of independent random variables and the stable distribution of order alpha $(0\le\alpha\le2)$. Show that $$\sum_{k=1}^{n}\frac{X_k}{n^{\frac{1}{\alpha }}}$$ if ${X_n}$ is ...
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Show That $Cov(X,\frac{1}{X})\le0$ if $X$ Is Positive Random Variable?

I Know That , If $X$ and $Y$ are independent then $Cov(X,Y)=0$ And $ Cov(X,Y)=E(XY)−µ_Xµ_Y .$ But I Don't Know How to prove The $<$ part And How to prove the part for positive random variable. Can ...
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Why the sum of two absolutely-continuous random variables isn't necessarily absolutely continuous?

Why "a sum of two absolutely-continuous random variables does not need to be absolutely continuous"? See problem 6.4 on page 6 in https://web.ma.utexas.edu/users/gordanz/notes/...
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Need to look at ordered samples for computing variance of sample mean in simple random sampling without replacement

In simple random sampling without replacement, the space of all possible unordered samples has cardinality $N \choose n$ and the space of all ordered samples has cardinality ${N \choose n} n!$. When ...
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Showing that $P(X_1>X_2) = \int_{0}^1 P(X_1>X_2 | X_2=x) f_{X_2}(x) dx$

I am going through this post in trying to prove the probabilistic interpretation of the AUC for a ROC Curve (for a classifier): The AUC for a ROC curve is the the probability of the classifier ...
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Binomial distribution vs. Discrete uniform distribution. vs. randomly selecting

I have an array of $n$ non-negative integers where each element can be randomly (uniformly) selected with repetitions from all the integers between 0 and $m$ (with $m>n$). So the probability of ...
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On random variables made up of independent random digits

Some random variables can be expressed as a binary expansion whose digits are chosen independently at random; this is called a convolution. One example of this kind of random variable is the one for ...
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Testing uniform distribution of a discrete function derived from a random number generator

(Forewarning: I'm a programmer, not a statistician, so I apologize in advance for any misuse of terminology!) I'm testing a known random number generator that implements the PCG algorithm. This RNG ...
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1answer
39 views

Can one uniformly generate complex numbers of absolute value less than a given constant $R \neq 1$? [duplicate]

Can one uniformly generate complex numbers of absolute value less than a given constant R? This would appear to be equivalent to picking points $(x,y)$ uniformly in a disk of radius R, where $x$ is ...
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Uncertainty principle in probability theory

In probability theory, there is the covariance inequality $$\operatorname{Var}(Y) \geq \frac{\operatorname{Cov}(Y,X)^{2}}{\operatorname {Var} (X)}.$$ In signal processing, there is a similar ...
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Technical name for difference between largest and second largest element in sequence

Let $X_1,\ldots,X_n$ be a sequence of $n$ real numbers (for example, iid random variables, etc.) and let $X_{(k)}$ be the value of the $k$th largest number, so that $X_{(1)} \ge X_{(2)} \ge \ldots \ge ...
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1answer
38 views

i.i.d assumption: formal definition vs. intuition [duplicate]

Intuition In ML, as I constantly run into the i.i.d assumption for datasets, I have an intuition of what this assumption really means. So if I'm not mistaken: "independent" means that ...
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If $X$ is a continuous random variable with support $A$, does this imply that the cdf of $X$ is strictly increasing on $A$?

If $X$ is a continuous random variable with support $A$, does this imply that the cdf of $X$ is strictly increasing on $A$? My guess is yes. But just in case, let me know if you can think of any ...
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Estimator of $Y$ in the simple linear regression model

In a statistics textbook I saw that the linear simple regression model is defined as \begin{equation} Y = \alpha + \beta x + e \end{equation} where $x$ is a value of the independent variable, $Y$ is ...
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Random number (between 0 & 1; > 5 decimal places) from skewed binomial/beta-like distribution, with set mean (same as mode) and set variance

Disclaimer: I already asked this question once, however it was misunderstood and I was told by a moderator to repost it: Random number (between 0 & 1; > 5 decimal places) from binomial/beta-...
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Conditions for gaussianity of sum of Gaussians [duplicate]

Under which conditions is the sum of a finite number of Gaussian random variables a Gaussian?
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What conditions are needed for $a_n = O_p(n^d) \implies E[a_n] = O(n^d)$?

Let $X_n$ be a uniformly integrable sequence of random variables. In a recent question I asked about the possibility of converting Big $O_p$ convergence in probability of the sequence $X_n$ to Big $O$ ...
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joint PDF of continuous and discrete random variables

Given exponential a random distribution X with PDF $f_X(x)=\lambda e^{-\lambda x}$ and a random variable Z with the PMF $p_Z[z]=0.5, z= \pm1$, I am trying to find the PDF of $Y=ZX$ (I also know that Z ...
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Can we go from $X_n = \mu + O_p(n^{-1})$ to $E[X_n] = \mu + O(n^{-1})$?

Let $X_n$ be a uniformly integrable (UI) sequence of random variables. If we have $$ X_n = \mu + O_p(n^{-1}), $$ then for $0 \le \delta < 1$ this implies $$ X_n = \mu + o_p(n^{-\delta}) \quad \quad ...
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Why does the superposition of two processes with geometrically distributed interarrival times not result in a process with similar distribution?

Assume we have two random variables $G_1$ and $G_2$ that both follow a geometric distribution of type B (so RVs can assume 0) with success probability $p = p_1 = p_2$ that describe the interarrival ...
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2answers
289 views

Random number (between 0 & 1; > 5 decimal places) from binomial/beta-like distribution, with set mean (same as mode & median) and set variance

Due to misunderstandings and as per request I have rollbacked this question to this previous state. Please do not answer this question instead answer this one: Random number (between 0 & 1; > 5 ...
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Calculate Conditional Expectation using dataset and covariance matrices

So I'm having trouble understanding what I'm doing wrong here. For context, I have some velocity components in my dataset for turbulence (simplified). I have flattened them out so my 3 velocity ...
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Probability distribution of training set t in Bishop's pattern recognition book

I'm currently struggling with understanding the Bayesian approach to machine learning. Which is one of the paradigms presented in Bishop Pattern Recognition and Machine Learning. Since some parameters ...
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1answer
60 views

Is $\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ uniformly integrable (UI)? What assumptions make it UI?

$\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ Let $\mathbf{X}_n$ be the usual data matrix in standard multiple regression where I have used the subscript $n$ to indicate the number ...
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When are these functions of a random variable independent?

Assume that $I$ is an indicator variable \begin{equation} I=\begin{cases} 1 &, \text{if} \,X<0 \\ 0 &, \text{else}\end{cases} \end{equation} and $X$ is a random variable. I want to know if $...
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1answer
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Separating $X$ from $Y$ in $E[(X^T Y))^p]$ for $p = 3$ and $4$?

Let $X$ and $Y$ be random vectors with $X$ independent of $Y$. When dealing with terms of the form $E[(X^T Y)^p]$ for $p \ge 1$, it is very useful to be able to separate the $X$ from the $Y$. For $p=1,...
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Is it possible to interchange the quantile operator and a measurable monotone function? $Q_\theta(f(X)) = f(Q_\theta(X))$

Let $Q_\theta(X)$ is the $\theta^{th}$ quantile of a random variable $X$, and if $f$ is a measurable strictly increasing function. I want to know if $Q_\theta(f(X)) = f(Q_\theta(X))$. I know that for ...
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Coskewness: three or two random variables?

The wikipedia for coskewness says coskewness is a measure of how much three random variables change together It then says If two random variables exhibit positive coskewness they will tend to ...
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Covariance between parameters from different regression models

Is there a formula for the covariance between regression slopes from different models, fitted to the same data? For example, if I have a finite and fixed sample, $S$, and models: $Y = b_1X$ $Y = b_2X ...
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Bounding the moments of the argmax of continuous process

I need to calculate/upper bound the second moment of the variable $t^{*} \triangleq \underset{t>\alpha}{argmax} \{W(t) - t^2\}$ where $W(t) \triangleq B(t) - B(t - \alpha), \alpha \in R^{+}$ and $...
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24 views

What's the term for a r.v. X “upper bounding” Y probabilistically? [duplicate]

What's the term for when a random variable $X$ has a higher probability of being greater than t than the probability of $Y$ being greater than $t$ for all real $t$? I recall reading a term for this ...
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1answer
88 views

Finding the probability that an exponential random variable is less than a uniform random variable

I have the following statement I am trying to solve: Let $T$~ $exp(2)$ and let $X$ ~Uniform$(0,1000)$ , where $T$ and $X$ are independent random variables, what is $P(500 + 1200T - X < 0)$ and ...
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1answer
48 views

Variable transformation under a discontinuous mapping function introduce different bias?

For any real number $x \in R$, consider its randomized version $y \sim \mathcal{N}(x,1)$. Now consider a mapping function of $y$ denote by $f(y)$. We want to study the following bias term $R(x) = | E[...
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1answer
52 views

How does a pdf change after a variable transformation with another random variable?

I have a probability density function of the energy $f(E)$ of a distribution of particles. Now, each energy gets shifted according to an angle $\theta$: $$E_{after} = E_{before} + g(E_{before}) \cos \...
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Strict Sense Cyclostationary and Shifting the X with $\theta$

Fellow stackexchangers, I did my best to put a topic that describes the question that I am going to ask. I am reading Probability, Random Variables, and Stochastic Processes by Papoulis and I am ...
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1answer
258 views

Show that maximum of two random variables is a random variable

If we have that $X$ and $Y$ are random variables, how do we prove that $Z=max(X,Y)$ is also a random variable ? I want to do this by showing that $Z$ is measurable, but I don't know how to do this.
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1answer
82 views

Random variable without pdf but with a cdf?

In this video, the professor says that some random variables have no pdf but do have a cdf. Also, in my course material, I studied that converging in mean was stronger than converging in cdf which ...
2
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1answer
33 views

When is Jensen's Inequality strict?

For a homework problem, I have to prove that for a random sample $X_1, \ldots, X_n$, drawn from a population with finite variance $\sigma^2$, with sample mean $\bar{x}$ and sample variance $s^2$, that ...

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