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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114 views

Correct or Not? Probability and Geometry [closed]

A stick of length $1$ is broken into two pieces of length $Y$ and $1−Y$ respectively, where $Y$ is uniformly distributed on $[0,1]$. Let $R$ be the ratio of the length of the shorter to the ...
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70 views

How to prove that this joint distribution is Gaussian without using probability densities?

Question: I am wondering if there was a way to prove this result without using probability densities: If $\bf x \sim \mathcal N (m, P)$ and $\bf y \;|\; x \sim \mathcal N (Hx, R)$, then $$\begin{...
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1answer
255 views

Correct or not? Mixed Bayes' Rule - Noisy Communication

In this problem, we study a simple noisy communication channel. Suppose that $X$ is a binary signal that takes value $−1$ and $1$ with equal probability. This signal $X$ is sent through a ...
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190 views

“A discrete random variable must have a finite range.” Is this true? Provide examples where possible [duplicate]

new to the community here and this topic on random variables. This was a discussion question posed to my group, and I have difficulties answering the question. I know that discrete means that its ...
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49 views

Variance of mean of correlated variables

Page 228 of THIS BOOK provides the formula for the variance of the mean of more than two correlated random variables: where $m$ is the number of variables, $r$ is the correlation between the ...
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65 views

Variance of difference of two correlated variables when working with random samples of each

I know that the variance of the difference of two correlated variables, $Y_1$ and $Y_2$ , is what the below formula shows, requiring $r$ the correlation between the two variables. But suppose I have ...
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130 views

UPDATED: Multiple lme models or MANOVA with random effects? Problem with singular fit

hope you can help me with this issue! In my study I have 4 outcome variables which correspond to the ratings I collected for 4 different psychological dimensions (liking, comfort, approach, ...
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373 views

Gamma Distribution for generating random numbers

I defined a gamma distribution with following parameters: shape,scale = 4.2503, 7037. This dribuation is used to generate random numbers. The random numbers will be recalculated to the x-asix value. ...
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358 views

When does the sum of the medians = the median of the sum

I have two random variables (say x1 and x2) defined by empirical probability distributions, and would like to calculate the median of their sum. Under what circumstances (in terms of the ...
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1answer
26 views

Proof that covariance of RV and group average is less than variance of RV

I have a random variable $X_i$ and a group of $N_j$ other random variables that includes $X_i$. Let's just call this group $J$. There are no distributional assumptions made on these RVs (other than ...
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1answer
29 views

Probability density function and distribution from samples to population

Heya got a small doubt here most of the discrete random variables utilize simple examples of situations like 3 coin toss or 2 dice rolls or girl/boys born. These examples are simple in the sense that ...
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19 views

Differences between matlab's PCA and theoretical Karhunen-Loève expansion

Matlab's pca offers coeff and score as results. Coeff represent the eigenvectors of the covariance matrix of the process in question, while score is the representation of the data in the components ...
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83 views

Probability density function for continuous random variable

The question might be very basic and stupid on certain levels, but please help me out here!! I recently picked up stats and went through discrete and continuous random variables. Discrete variables ...
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63 views

Interpretation of the technical requirement on a random variable

I found a slide where there is the definition of a random variable and after a technical requirement difficult to understand for me. Can you explain it by using a counterexample please? What happens ...
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210 views

Are the errors in this formulation of the simple linear regression model random variables?

On page 21 of Applied Linear Regression, fourth edition, by Sanford Weisberg, the error $e_i$ for case $i$ under the simple linear regression model is defined to be $y_i - E(Y | X = x_i)$, where $E(Y |...
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102 views

Sum of a random number of r.v.'s [closed]

A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, ...
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Does $P(X>x, Y>x)= P(X>x)P(Y>y)$ implies independence?

We know, by definition, that two random variables are independent if $$P(X\leq x, Y\leq y)= P(X\leq x)P(Y\leq y).$$ If, insted, I have that $$P(X>x, Y>y)= P(X>x)P(Y>y),$$ does this also ...
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25 views

Transformation of a random variable and KDEs — when is reweighting needed?

Suppose I have been given some data $X$ that I wish to resample according to their empirical distribution. For whatever reason, I decided to transform these variables to some other space $Y = f(X)$ ...
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43 views

Finding the expectation: Draws to get 2 gold coins

Omar has saved 11 precious coins, 7 of which made of gold, in a jar. He draws the coins one by one from the jar, to find the gold coins. If $X$ is the number of coins drawn until he has found the ...
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72 views

Meta-analysis with random effects on prevalence data with > 2 possible outcomes: is it necessary that the weighted prevalences all add up to 100?

I am running a meta analysis on prevalence data. Participants can belong to one of four groups. Each study produces the percentage of participants that belong to each group (e.g., Group A: 50%; Group ...
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1answer
82 views

An unreplicated nested design with repeated measurements: looking for a proper analysis

I want to analyse a data from a study which was not designed properly due to logistic reasons. There are 3 plots, each has unique management type (see figure below). At each of these plots several ...
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58 views

Probability of an event at the outcome of a binary source

Let a binary (and independent) source $S$ generating a binary sequence with the following probabilities: $p$ for the symbol '0' and $1-p$ for the symbol '1'. What is the probability that $S$ gives $n$...
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103 views

What is the Markov blanket of a deterministic variable?

The following Bayesian network contains a node which is deterministically dependent on its parents: the variable $either$ is simply the $OR$ function of its parents $tub$ and $lung$. By the graph, ...
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54 views

Is the concept of a random variable useful or necessary in machine learning?

I apologize in advance if this sounds like a really bad question. Probability and statistics is extensively used in machine learning, and random variable is extensively used in probability and ...
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151 views

Probability of an event vs probability of random variable

How does the probability of an event differ from the probability of a random variable? For example, consider the event where there are 4 babies born. Sample space: ...
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33 views

Is the following definintion of Discrete Random Variable given in the book of Joe Blitzstein correct?

The following is the definition given in the book: A random variable $X$ is said to be discrete if there is a finite list of values $a_1, a_2,\ldots, a_n$ or an infinite list of values $ a_1, a_2,...
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16 views

What does this assumption mean regarding equal marginal densities?

Suppose that we have a random variable $\epsilon$ with density $q(\epsilon)$ and $w = t(\theta, \epsilon)$, where $t$ is a deterministic function of a constant $\theta$ and random variable $\epsilon$. ...
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135 views

Convergence in probability does not imply convergence in $r^{th}$ mean

I am confused regarding convergence in probability and convergence in $r^{th}$ mean. I am able to prove that convergence in $r^{th}$ mean implies convergence in probability, which is not true. Let me ...
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72 views

Changing a conditional probability to a deterministic function

Suppose that we have a conditional density function $p(y|x;\theta^*)$, where $\theta^*$ represents distribution parameters and are assumed to be deterministic. Is it possible that we write this ...
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54 views

Bayesian estimation from sum of two random variables

Let's say I have a set of observations $Y=\{Y_1,\ldots,Y_N\}$ where each observation is created as the sum of two random variables, i.e. $Y_i=X_{1,i}+X_{2,i}$. Also, I know that $X_1 \sim Dist_1(\...
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1answer
53 views

A simple question from ANOVA

As per definition of $F$ statistic, $F= \frac{MST}{MSE}$ where MST and MSE denote mean square due to treatment and error respectively. From this definition of $F$ am I right in saying that $F$ ...
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63 views

Getting sets of random correlated variables

For the training of a machine learning model I need to add additional features, and these features are correlated. I need to run the model N times adding these features with random values, and for ...
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3answers
76 views

Does Normality Imply Randomness?

I have data indicating the number of counts per minute (so 60 rows in total - one for each minute - and # of events in that minute). I have ran the Shapiro - Wilk test which implies the data does not ...
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12 views

Permuting RV order in stacked Auto-regressive Flows for density estimation

Brief background: Normalizing flows such as detailed in MAF and B-NAF use an auto-regressive formulation such that highly expressive bijective transformations of the RVs satisfy the probability chain ...
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69 views

Difference between tight and uniformly tight random variables?

This wikipedia page implicitly says that “tight” and “uniformly tight” random variables refers to the same concept. I find this somewhat surprising. Are there contexts in which a distinction is made ...
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23 views

Where is the single-crossing point between a distribution and its single mean-preserving spread?

Consider two random variables, $Z_A$ and $Z_B$, with the same expected value $\mu$, having distribution functions $F_A$ and $F_B$, respectively. Let $Z_B$ be obtained by a single mean-preserving ...
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32 views

I.i.d.-ness of some functions of random variables

I have some doubts on the i.i.d.'ness of some functions of random variables. The framework What I'm describing below is a simplied version of a well known model in economics of demand and supply. ...
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69 views

Understanding simplification of constants in derivation of variance of regression coefficient

In looking over TooTone's answer in Derive Variance of regression coefficient in simple linear regression, there's a step in line 3 below where $(\beta_0 + \beta_1x_i + u_i )$ is simplified to $u_i$ ...
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1answer
95 views

Is the expected value of a RV same as the mean of the corresponding pdf?

As we know the expectation of a RV $X$ or a function, say $g(X)$, of $X$, both with pdf $p_{X}(x)$ is $$ \begin{array}{*{20}{c}} {X \sim {p_X}(x):}&{E[X] = \int {x.{p_X}(x)dx} }\\ {g(X) \sim {p_X}...
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29 views

Can this be simplified $\mathbb{E}_{q(\vec{z} \mid \vec{x})}\left[ \log {p(\vec{x} \mid \vec{z})}\right]$?

Assume that $p$ and $q$ are two distributions and $x$ and $z$ are two random variables. Can the following term (which appears in the paper Auto-Encoding Variational Bayes) be further simplified? $$\...
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27 views

random factos really significant?

I have some confusion about random factors inclusion or not. I've used the function glmer.nb of the library MASS to analyse the effects of two fixed factors (temperature: 2 levels and salinity:3 ...
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48 views

Probability of getting a negative mark if one chooses options randomly in a MCQ question with negative marking

In a test with 100 multiple choice questions, a student chooses to pick an option randomly. There are four options for each questions. Answering a question correctly will give the student 1 mark, ...
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46 views

Decomposing a random variable into marginals and copula

I’m having trouble getting understanding how to actual construct a copula, from my understanding it captures the purely joint features of a joint distribution. I’ve been working with the following ...
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1answer
35 views

Probability of having an increasing trend in normal variates

Let $x\sim N(\mu,\sigma)$ and $x_i$ is ordered instances of random variate of $x$ for $i=1...n$. What is the probability that the series is in increasing (or decreasing) order? The problem is finding ...
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29 views

Perpendicularity of random variables?

I am reading Bechavod et al. (2017) [1], and at page 3 there is written: In the example, each data point lies in $X = (X_1,X_2) = \{0, 1\}^2$ and has two features—$X_1 = A$ is the protected ...
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50 views

Supremum of parameterized random variables over compact set

Suppose that we have a parameterized real-valued discrete stochastic process $x(t) :=\{x_k(t)\}_{k=1}^\infty$, such that $t$ assumes values in a compact set $T\subset \mathbb{R}^d$ for some finite ...
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38 views

Can two random variables be independent in some basis and dependant in other?

If some random variables forming N dimensions are dependant on each other is it possible that in a different coordinate system they'd be independent? For example if (X, Y) are two dependent RVs is it ...
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29 views

Decomposition random variable with conditional expectation [closed]

Why given some information set $I$, any random variable $x_t$ can be decomposed into the sum: $$x_t = E(x_t | I) + v_t $$ where $E(v_t | I) = 0$. I'm looking for a clear proof.
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21 views

Poisson and Gamma distribution for testing randomness

In genetics I want to test whether InDel (insertion and deletion in DNA) sizes occurs with the same probability. I heard that I should gamma distribution to model it. I found ...
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172 views

What is the distribution of the difference of two iid noncentral Student t variates

Let $X_1$ and $X_2$ be iid non-central t random variables. I'm interested in the question: what is the distribution of $X_1 - X_2$? i.e. what is the distribution of the difference of two iid ...

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