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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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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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31 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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27 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
36 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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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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33 views

Columns vs Random Variables [closed]

A random variable, $X$, is an event space $\Omega$, along with a map $f: \Omega \rightarrow \mathbb{R}$. The map $f$, assigns probabilities to elements of the event space. We have a notion of mutual ...
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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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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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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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2answers
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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
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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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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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16 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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1answer
16 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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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
26 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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26 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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1answer
35 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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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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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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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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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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How do I generate a random variable whose CDF is the product of given CDFs?

I am looking at a finite set of independent random variables ${\left\{ {{X_i}} \right\}_{1 \le i \le n}}$ with CDF ${F_i}$, with the aim of generating a random variable such that it has the CDF ${G_1}(...
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1answer
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Probability density functions

The probability density function of X is defined by: f(x) = α 0 ≤ x ≤ 1 β(x-4)^2 1 ≤ x ≤ 4 Show that the exact values of α and β are 1/2 and 1/18 respectively: We know that the total ...
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How to write $P(\delta = 1 | X = x)$ as function of $P(\delta = 1 | X = x, Y = y)$

Suppose that $\delta$ is a Bernoulli random variable, and suppose that $X, Y$ are continuous random variables. Is there a way to write $P(\delta = 1 | X = x)$ as function of $P(\delta = 1 | X = x, Y ...
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1answer
52 views

Summation of cosine of uniform random variable

I read that the PDF of the sum of cosines of a random variable, which is uniformly distributed, is a non uniform distribution; something like inverse square root of the random variable. My doubt is, ...
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1answer
42 views

Compute $E(X_1|X_1+X_2)$ $X_1, X_2$ both iid $Exponential(1)$

I recently stumbled across this question on CV: Conditional expectation conditional on exponential random variable And really liked the answer provided by @Rush, but I wanted to try to compute this ...
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1answer
32 views

How to interpret estimates and correlation of random effects (intercepts and slope) in a mixed-effects model in a Bayesian framework(brms)?

I do not understand how to interpret random slopes from the output of brms Among others, I read this post on the output from ...
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1answer
27 views

Why can we expand terms with random variables in the variance formula?

$\newcommand{\E}{\mathrm{E}}$$\newcommand{\Var}{\mathrm{Var}}$In the proof for showing the alternative formula for variance, i.e. $$\E[(X - \mu)^2] = E[X^2] - E[X]^2$$ I typically see the following ...
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Inferential Testing

I am currently working on my dissertation, which is investigating the relationship between working memory capacity and problem-solving ability. I have two independent controls task 1 and task 2, both ...
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What does the integral of a function times a function of a random variable represent, conceptually?

I am trying to understand conceptually what does the following give me or tell me: $$\int f(x) \cdot g(x) \, dx$$ where $f(x)$ is any continuous function of $x$ and $g(x)$ is the probability ...
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is it reasonable to consider the possible outputs of tossing 2 coins 10 times as a bivariate distribution?

a bivariate distribution involves a pair of random variables (x, y). in an experiment of tossing 2 fair coins 10 times. let x denote the possible outputs of tossing the 1st coin 10 times. let y denote ...
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1answer
21 views

Expectation of a sequence of gaussian variables

Suppose we are given a sequence of iid Gaussian random variables $\{x_k \}$ with zero mean and unit variance. We create a new random variable $$X_n = e^{\sum_{n\ge k}x_k}.$$ How does one go about ...
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Random variables in machine learning

In their text on machine learning, Ben-David and Shalev-Shawrtz (see https://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf) develop a ...
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Distribution of sums and differences of n correlated normal random variables

Suppose $x_1\sim\mathcal N(2,0.5),x_2\sim \mathcal N(2,3),$ and $x_3\sim \mathcal N(2.5,7)$ with correlations $\rho_{(1,2)}=0.3,\rho_{(1,3)}=0.1,$and $\rho_{(2,3)}=0.4.$ What is the distribution of ...
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The joint distribution of Y=AX and Z=BX given a projection matrix A and residual maker matrix B, and a random vector X with known pdf?

This question follows on from a previous question I asked which was answered. It turns out my question lacked some important details, which was revealed by the answer posted on that thread. This is ...
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2answers
72 views

Probability mass function for the first non-increasing sample from a random sequence

Consider a sequence of random numbers drawn IID from some distribution $g(x)$. How would I determine the distribution of the value of the first sample from that sequence which is not greater than all ...
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1answer
57 views

What does the pmf of a discrete random variable look like if it can take on the value $\infty$?

What does the pmf of a discrete random variable look like if it can take on the value $\infty$? Consider a stopping time $\tau$ of a Markov chain, which is a random variable that takes values in the ...
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1answer
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How to derive the joint distribution of Y=AX and Z=BX given a random vector X with known pdf?

Given a random vector $X \in \mathbb{R}^k$, with a known pdf given by $f_X$. If $Y, Z \in \mathbb{R}^k$ are defined by $Y = AX$, $Z = BX$, where $A,B \in \mathbb{R}^{k\times k}$ are different, given, ...
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1answer
36 views

Couldn't understand conditional distribution and density function

The random variable X has a distribution function as shown in the graph below and $ Y=X^2$. Find a. $P(1/2 ≤ X ≤ 3/2)$ b. $P(1/2 ≤ X ≤ 3/2 | Y ≤ 1)$ g. $P(X+Y ≤ 3/4)$ I couldn't understand how ...
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1answer
23 views

How can I compute expected return time of a state in a Markov Chain?

I was watching a YouTube video regarding the calculation of expected return time of a Markov Chain. https://www.youtube.com/watch?v=X_Ll0-Ytu7U&vl=en I haven't understood the calculation of $m_{...
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Mean and Variance of weighted sum of n random variables? [duplicate]

Suppose we have n jointly distributed random variables $x_i,i=1,...,n,$ with mean and variance $E(x_i)=\mu_i$, $Var(x_i)=\sigma^2_i$ and covariance $Cov(x_i,x_j)=\sigma_{ij}.$ Then the weighted sum of ...
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Product of a linear function of IDD variables [duplicate]

if $r_i$ are IID variables with $E[r_i]=μ$ and $f$ is a constant, is the following correct? $E\left[\prod_{i=1}^n(1+f\space r_i)\right] = \prod_{i=1}^nE\left[1+f\space r_i\right] = \prod_{i=1}^n(1+...
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Sufficient condition for existence of maximal/optimal coupling

Let us consider two random variables $X$ and $X'$ defined on the same measurable space $(E, \xi)$. Is there a sufficient condition on the distributions of the random variables $X$ and $X'$ for ...
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1answer
16 views

What is the difference between multivariate random variables and sample random variables? [closed]

Multivariate random variables consists of more than one random variable which may be independent , eg. Height , weight , age can be called three random variables and we can write their joint ...
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19 views

Transformation of random variables and Jacobian

When transforming 2+ continuous random variables, you use a Jacobian matrix and compute the determinant. Do you also compute the Jacobian for discrete random variables?
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1answer
28 views

How to normalize mutual information between to real-valued random variables?

How can I normalize mutual information between to real-valued random variables using Python or R? ...
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31 views

Distribution of N objects into C bins that are then sorted?

Let's say we have $C$ bins and $N$ indistinguishable objects. For each object we choose one bin at random where each bin is equally likely (with probability $1/C$). Let $B_k$ be the number of objects ...
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2answers
182 views

What does it exactly mean if a random variable follows a distribution

Imagine there's a random variable such as $ε$. Then we say that $ε$ is i.i.d and follows a normal distribution with mean $0$ and variance $σ^2$. What does this mean? Is this not a variable anymore? ...
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1answer
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How can observations of random variables be IID, if they are not themselves random variables?

Suppose we perform some experiment which results in an outcome $\omega \in \Omega$. A random variable $X$ maps $\omega$ to a real number, and the (discrete) distribution $P(X)$ maps $X$ to $[0, 1]$. ...