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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If a random variable is independent of two random variables separately, is it independent of them jointly? [duplicate]

Does the following rule hold? $\left\{ \begin{array}\\ X \perp Y \\ X \perp Z \end{array} \right. \Rightarrow X \perp Y,Z $ If it does not hold, what is a counterexample?
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What is the expectation of the Cholesky factor of a Wishart distributed random matrix?

Let a $d-\text{dimensional}$ Wishart random variable with $\nu$ degrees of freedom $\Sigma$ be distributed according to $\mathcal{W}(\Sigma|\Sigma_0, \nu) \propto |\Sigma|^\frac{\nu-d-1}2\exp{(-\...
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Expressing a continuous random variable in terms of discrete samples

I am attempting to capture what I am doing when running a simulation, drawing discrete random samples from a normal distribution. Assume $X \sim \mathcal N(\mu,\sigma^2)$. I then draw $n$ samples $S_i$...
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How can I calculate the probability that one random variable is bigger than a second one?

I have five random variables which are independent and each one of them has a continuous uniform distribution on the interval $ [0,2]:$ $$X_i = \operatorname{Uniform}[0,2].$$ I want to calculate the ...
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1answer
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Confusion in definition of independent and identically distributed random variables [duplicate]

From what I learnt, a random variable is a function which assigns real values to outcome space, and the probability distribution is a function that assigns probability to different values produced by ...
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Discrete and Continuous variables. What is the definition?

The definition of a continuous variable in our class seems to be, well, not a definition, as there are exceptions not included in its definition. I am a 4th year math student and find it appalling ...
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Stochastic simulation, what to do after generate the initial random sample

I don't have a background in statistics but currently learning the basics. I want to do a stochastic simulation, which I assume here I should iterate my simulation multiple times. And I am stuck now ...
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Hypergeometric distribution with no number of defect items information

So I am looking at the problem here: A hospital has received a shipment of 25 new X-ray machines. The hospital could lose its license for housing X-ray machines if the machines are not properly ...
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Why I cannot generate random numbers having a truncated lognormal distribution?

My deduction is: When the distribution is truncated, a normalization factor should be introduced: \begin{equation} g(x) = \frac{C}{x\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{\ln{x}-\mu}{\sigma}\...
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Binomial probabilities under different sample sizes

I have a question about the binomial distribution probabilities, more like a curiosity that I couldn't prove. Let $n_1<n_2$ and let $X_1\sim Bin(n_1,0.5), X_2\sim Bin(n_2,0.5)$. Suppose that $n_1,...
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Random Variable Confusion

What is the difference between saying X is a random variable following normal distribution and X is normally distributed. Is the random variable implicit or is X in the second case some other entity?
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How can you estimate the actual number of options when there is some "fuzziness" involved? (Valve check/parking tickets)

I received a parking ticket the other day. By checking the valve positions the parking company claim that my car hadn't moved for a certain period. The protocol from the valve check shows that the ...
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Largest order statistics of non-identical distributions when extra information is available

Suppose we have two independent draws, one from a distribution $F_1$ and the other from a distribution $F_2$. The two distributions have the same support, say $[0, 1]$. The distribution of the largest ...
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1answer
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What is the distribution of the difference of two independent multinomial random variables?

Say I have two independent random vectors $X_c$ and $X_f$. The random vector $X_c$ is composed by three random variables: $X_{1c}$, $X_{2c}$ and $X_{3c}$. The second random vector $X_f$ is composed by ...
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Expectation of number of voters who agrees with the proposed plan under the majority voting

This question raised in my research work and it is quite challenging for me. Consider a committee $D$ of $n$ voters, among these $n$ voters there are $n_1$ voters agreeing with the proposed plan, ...
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Find the probability density function of the joint pdf $2(x+y)$

Let $X$ and $Y$ be random variables for which the joint probability density function (p.d.f) is as follows: $f(x,y) = \binom{2(x+y) \space \space for \space \space 0\le x \le y \le 1}{0 \space \space \...
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Calclulating the amount of duplication in a pool of random numbers [duplicate]

I want to figure out a way to determine how many unique numbers there would be in a given pool of random numbers. For example, lets say I generate 20 random numbers, each with a value between 1 and 20....
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Covariance of some random variables

I am given 2n-1 random variables, namely X₁, X₂... Xₙ, Xₙ₊₁... X₂ₙ₋₁. I also have E(Xᵢ)=𝜇 and Var(Xᵢ)=𝜎² for i=1,2,...2n-1. Suppose Y=X₁+X₂+...+Xₙ and W=Xₙ+Xₙ₊₁+...+X₂ₙ₋₁ and I am asked to calculate ...
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1answer
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Variance of discrete distribution exceeds variance of discrete uniform distribution

I am not a mathematician, so I don't quite understand how comes that a variance of some discrete probability distribution could exceed the variance of the discrete uniform distribution. I thought that ...
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Regression with random input variables

Take a black-box model which is expensive to evaluate. Say you have $N$ observations of this black-box model denoted $\{Y_i\}_{i=1,\ldots,N}$ at the points $\{X_i\}_{i=1,\ldots,N}$. We take an input ...
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1answer
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How to generate random points from a custom curve? [closed]

my question is how to generate random points ($\theta_1,\phi_1$) on a curve determined by : $ \arccos\left(\cos(\theta_1)cos(\dfrac{\pi}{6})+\sin(\theta_1)\sin(\dfrac{\pi}{6})\cos(\phi_1)\right) = \...
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Find the probability density function of $Y = X_1 + X_2$ [duplicate]

Suppose that $X_1$ and $X_2$ are i.i.d random variables and that each of them has a uniform distribution on the interval [0,1]. Find the probability density function $Y = X_1 + X_2$ I understand that ...
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How to construct the confidence band for a timebased sample of n individuals

Let $X^{(i)}(t)$ be some measurement at time $t$ for individual $i$. Let $\bar{X}(t) = \frac{1}{n} \sum_{i=1}^n X^{(i)}(t)$ be the mean of the individuals at time $t$. Let $\bar{X}_{t_0} = \frac{1}{T}\...
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The distribution of $\frac{nX_{(1)}}{\sum_{j=1}^nX_j}$, where $X_1,...,X_n$ are iid from an exponential distribution

Suppose $X_1,...,X_n$ are iid from a continuous distribution with pdf $$f(x) = \lambda e^{-\lambda(x-\theta)},\:\: x>\theta,\: \theta \in \mathbb{R}$$ What is the distribution of $\frac{nX_{(1)}}{\...
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When you are running an analysis with only an intercept, does it make sense to include random subject and item intercepts?

Lets image a study where people get two shapes and are told to pick one. They each get 40 trials, each with a different shape/colour. There are 40 different pairs of shapes. There are 100 participants....
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When was a random variable first called a "random variable"? And why is it called as such?

From measure theoretic foundations, it is clear that a random variable is neither random nor a variable. It is a deterministic function developed as follows: Construct probability space: $(\Omega, \...
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Expectation of a scaled random variable

Suppose we have 2 continuous random variables $X$ and $Y$, where $Y = cX$ for a $c \in \mathbb{R}$. The probability density of $X$ range from $-\infty$ to $\infty$. I know that the expectation values ...
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Same random seed for different dependent variables in machine learning?

Consider dependent variables $Y_1, ..., Y_n$, some of which may be correlated. For each dependent variable, I want to use machine learning with sample splits for model selection. Clearly, for each ...
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Probability that any element of a random unit-length vector is large [closed]

Given a vector $X \in R^n = \{x_1, x_2, ..., x_n\}$ drawn uniformly such that: $x_i \in [0, 1]$ for all $i$; and $\sum x_i = 1$, how would you find the probability that any of the $x_i > y$, for ...
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Significant bias introduced into simple simulation

Introduction Service is allocated to an infinite source of customers i.e. there is always a service in progress. The duration of the $i^{th}$ service is generally distributed $\Delta_i \sim F_{\Delta}$...
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1answer
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What is $ \frac{d \ E(ln(y)|X)}{d \ y}$ in OLS?

Assume that the true model (DGP) is $ ln(y) \ = \ \beta_0 \ + \ \beta_1 ln(x_1) \ + \ \cdots \ + \ \beta_k x_k \ + \ \varepsilon \hspace{3em} \text{where } \ \begin{bmatrix} x_1\\ \vdots\\ x_k\\ \...
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Existence of a random vector whose distribution has full support

Consider a $6\times 1$ continuous random vector $$ \eta\equiv (\eta_1,\eta_2,..., \eta_6) $$ satisfying the following property: $$ \underbrace{\begin{pmatrix} \eta_1\\ \eta_2\\ \eta_3 \end{pmatrix}}_{\...
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Simplify Equation with Random Variables

I'm wondering if whether the following problem has a solution. Suppose we have i random variables, all independent, and all following a Bernoulli distribution with parameter $p_i$ (all $p_i$'s are ...
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Criteria to be called a random variable?

I've read that to call something a random variable, that thing must be the result of a statistical experiment. So it got me thinking in which situations might we have an actual bias? For example, ...
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Infimum of a random function

Suppose that we have a random variable $X$ and this is a function of an index $t\in T$. Here, what is the meaning of the infimum of this random function? $$\inf_{t\in T}X(t)$$ How can we interpret ...
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1answer
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visualization of posterior distribution for a matrix?

Via Bayesian analysis, I generate the posterior samples for a 2D matrix, representing a correlation matrix for instance. Is there any way or tool to visualize these two dimensional random samples?
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How can I generate correlated Pearson III random variates

I would like to generate correlated Pearson III random variates. Any suggestions would be most welcome. I can generate independent Pearson III random variates using the rlmomco function in the lmomco ...
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How to determine the likelihood a random number generator is using a uniform distribution?

Let's say I have a blackbox function generate_number() that generates a random number between 1-N; and assume N is known. Each ...
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3answers
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If $20 $ random numbers are selected independently from the interval $(0,1) $ probability that the sum of these numbers is at least $8$? [closed]

If $20 $ random numbers are selected independently from the interval $(0,1) $ what is the probability that the sum of these numbers is at least $8$? I tried to take this question https://math....
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1answer
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Bernoulli distribution

Definition 3.3.1 (Bernoulli distribution). An r.v. X is said to have the Bernoulli distribution with parameter p if P(X = 1) = p and P(X = 0) = 1 − p, where 0 < p < 1. We write this as X ~ Bern(...
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Assigning Random Variables

(Sum of die rolls). We roll two fair 6-sided dice. Let T = X + Y be the total of the two rolls, where X and Y are the individual rolls. The sample space of this experiment has 36 equally likely ...
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Euclidean Norm normalized Normal Distribution

Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anistropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different. ...
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1answer
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Intuition about the definition of random variables? [duplicate]

I have been struggling a bit with the definition of random variables, as it seems to me a bit of an amorphous concept connected to quite a few different ideas. The Wikipedia article also didn't clear ...
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If I consider the fixed factor as a random slope, the p-value changes from p<0,05 to p>0,05

I'm having a hard time trying to understand the differences between these two models and why the first one shows correlation (p-value < 0,05) but the other one doesn´t (p-value > 0,05). I would ...
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1answer
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Probability of not picking a row in a random draw where the number of rows are N

There are $N$ rows :$R_1, R_2,R_3,..., R_N$. What is the probability of not picking a row in a random draw? My try and understanding : Let $X$ be a random variable which is defined as follows: $$X = \...
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1answer
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Finding the Maximum Likelihood Estimator for a random variable of mixed type

I am having issues solving the follow problem from my textbook: Suppose we have $x_1,...,x_n$ i.d.d. data from a r.v. $X$ with unknown distribution function $F_\theta$, for $\theta = (\theta_1,\...
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Mean Sum of squared Error in TWO WAY ANOVA

This is a randomised block design with each b blocks and m conditions. Each block-condition group has n data points making a total of $n\times m \times b $ data points $Y_{ijk} = \mu + T_j+w_k+\...
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Transformation of Random variable

Suppose I have $N$ random variables $\{X_i\}_{i=1}^N$, and I want to calculate the expectation of a function of these random variables: $$\mathbb{E}[g(X_1,\cdots,X_N)h(X_1,\cdots,X_J)]=\int g(X_{1},\...
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1answer
85 views

How to test for germination differences with 20 replicates of 3 seeds per treatment group?

I need to test for differences in germination success between treatment groups, however, instead of all seeds being independently sown, I have 20 growth tubes with 3 seeds in each per treatment group (...
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2answers
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How do I generate $n$ random variables that follow a correlation matrix with individually log normal distributions?

Short and sweet: I'd like to model $n$ random variables representing price changes of individual assets. Each of these should be distributed as a log normal variable with a median of 1. Is there a way ...

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