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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Problem on convergence of sequence of random variables

Given any sequence of random variables $\{ X_n \}$; how do I show that there exists a sequence of real numbers $\{ \alpha_n \}$, such that $\{ \alpha_n X_n \}$ converges in probability to 0 ?
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72 views

Is it okay to write the square of expectation of a random variable $X$ as $\mathbb{E}^2(X)$?

Is this notation accepted when I write $\text{Var}(X)=\mathbb{E}(X^2)-\mathbb{E}^2(X)$?
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47 views

Probability than empirical mean of one binomial RV smaller than another

Lets suppose I have two binomial random variables: $X \sim B(n_1, p_1)$ and $Y \sim B(n_2, p_2)$. I would like to calculate the probability than the empirical mean of $X$ be smaller than the empirical ...
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1answer
59 views

Derived Distribution from normal distribution

\begin{align} X_{1} \sim N(\mu_{1} , \, \sigma_{1}^2 ) \\ X_{2} \sim N(\mu_{2} , \, \sigma_{2}^2 ) \end{align} Assume $X_{1}$ and $X_{2}$ are independent, what is the distribution of $ Y = 1/X_{1} ...
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52 views

Mixed Distribution Question

Let $$f\left(x,y\right)= \begin{cases} \frac{1}{3}, & x = 1, \, 0 \le y \le 1 \\ \frac{1}{6}, & x = 2, \, 0 \le y \le 2 \\ \frac{1}{9}, & x = 3, \, 0 \le y \le 3 \end{cases}$$ I want to ...
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90 views

Figuring out the bounds of integration over a joint pdf

I have two random variables $X$ and $Y$ where the support of $X$ and $Y$ are the following: $0\leq X\leq 1$ and $0\leq Y\leq 1$. I also have their joint distribution, i.e., $f_{X,Y}(x,y)$ and I want ...
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15 views

Nested GLMMs: which are my random factors?

I am analyzing the number of seed capsules between different genotypes (A,B and C) I have 4 replicates for each genotype and in each of these replicates, I have 8 plants. Here is an example of the ...
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19 views

Implementation of sequence of binary random variables in r

I am trying to implement a random variable in R, and I want to generate a sample from it. The random variable looks like this: we have $P(X_{n}(\omega)=\frac{n}{n+1})=0.5$ and $P(X_{n}(\omega)=(-1)^{...
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1answer
47 views

Difference between power of a random variable and product of random variable with itself

In the R package distr there are two infix operators ^ and ...
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521 views

Is the sum of a discrete and a continuous random variable continuous or mixed?

If $X$ is a discrete and $Y$ is a continuous random variable then what can we say about the distribution of $X+Y$? Is it continuous or is it mixed? What about the product $XY$?
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61 views

Distribution of the ratio of two shifted generalized gamma random variable

$X \sim \mathrm{GG}\left(p,d,\theta_{1},\mu\right)$ where $p$ is power, $d$ is shape, $\theta_1$ is scale and $\mu$ is location parameter. Also Consider $Y \sim \mathrm{GG}\left(p,d,\theta_{2},\mu\...
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17 views

Expected value / variance of continuous random var in terms of another

Let $X_0, X_1, ... , X_n$ be independent random variables, each with mean $\mu$ and variance $\sigma^2$. Let $Y_i = X_i − X_{i−1}, i = 1, 2, ... , n.$ What is the expected value of $\bar{Y}$ What is ...
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16 views

Maximum of two random variables

Say I here are two operations that start simultaneously Operation x has a mean of 5 days and a standard deviation of 3 days Operation y has a mean of 6 days and a standard deviation of 2 days I am ...
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1answer
21 views

Self study - Determining function codomain to study convergence

I'm having a serious problem in and old exam paper, specifically a question on convergence of random variables. Let $X$ and $Y$ be two i.i.d exponential random variables with parameter $\lambda$, so ...
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1answer
22 views

If X is Beta distributed, what is distribution of Y: Y=1 for X>0.5 and Y=0 otherwise

Consider random variable $X \sim \mathrm{Beta}(\alpha,\beta)$ What is the distribution of $Y$ defined by $Y\in \{0,1\}$, $Y=1$ if $X>0.5$ and $Y=0$ otherwise.
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Related fixed and random factors in linear mixed-effect models and posthoc tests

I am trying to use mixed-effect modeling to analyze brain wave data from two groups of participants when they were presented with two distinct stimulus. The data points (scalp voltage) were gathered ...
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55 views

Generate random variables with negative binomial distribution in R [closed]

How do I create a function in R in order to generate "n" random variables with a negative binomial distribution? This is for homework, so rnbinom doesn't help.
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7 views

Create a function to generate negative binomial random variables in R [migrated]

This is for homework, so rnbinom doesn't help I've created two functions: combr (combinations with repetitions) and another called negbinom. This is combr: ...
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19 views

Why are these two samples correlated with each other (ISLR package)?

On pg. 44 of Introduction to Statistical Learning: Here we create two correlated sets of numbers, x and y, and use the <...
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16 views

Random right cenoring

I observe a series of values from different trials ($Y_1, ... Y_N$). All values come from the same distribution ($F(\cdot)$). Trials do not have the same number of values ($N$ differs across trials)....
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1answer
17 views

Minimizing MMSE over positive random variables

Let X be a random variable with a finite second moment. We know that: Argmin E(X-Y)^2 = E(X|g), Where the minimum is taken over all g-measurable random variables Y. How can I find argmin E(X-Y)^2 ...
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5 views

Why is the parameter and the random variable swapped in this conjugate distribution pdf?

I'm reading a journal titled Claims reserving in the hierarchical generalized linear model (hglm) by Gigante, Picech and Sigalotti. In the distributional assumption for the unobserved risk parameters, ...
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358 views

Is the sum of a large number of independent Cauchy random variables Normal?

By Central Limit Theorem, the probability density function of the the sum of a large independent random variables tends to a Normal. Therefore can we say that the sum of a large number of independent ...
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3answers
43 views

Prove that this doesn't converge almost sure to 0

Suppose we have $X_n$ a random variable, that can take two values: $X_n = \begin{cases} 0, & \text{with probability 1 - $\frac{1}{2n}$,} \\ n, & \text{with probability $\frac{1}{2n}$} \end{...
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17 views

Treating X as non-random under random sampling

Random sampling allows us to treat values of independent variables that have been sampled as non-random (for the purpose of proving the unbiasedness of OLS estimators). How is this possible? My ...
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1answer
25 views

Transformation from skewed to symmetric distribution

Let us consider a positive valued random variable $X$ which is following a positively skewed probability distribution. Is it possible to a get a function $f$ (one-to-one) for which $f(X)$ follow a ...
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1answer
57 views

Exemple of real-life non-linear correlation in time series?

I am looking for a dataset of 'real-life' time series that exhibits non-comonotonic / non-countermonotonic dependence. I am not looking for the textbook X^2 correlation, but interesting yet real non-...
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29 views

Difference between (quantitative,qualitative) vs (discrete,continuous)

Could someone please clarify the difference between these seemingly interchangeable groups? (quantitative,qualitative) vs (discrete,continuous). Obviously they are not, but I cannot seem to pinpoint ...
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8 views

Calculating joint entropy of two variables having null values

I calculated the joint entropy of two random discrete variables by zipping their values in a 2-tuple and applying the formula: where n is the number of distinct classes and q is the probability of ...
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3answers
36 views

Square of the Sample Mean as estimator of the variance

Suppose we have the following random variables $X_1$, $X_2$,....$X_n$,.., that are $iid$ but we dont know what distribution they follow. I know that the sample mean $\bar{X}$ is an unbiased ...
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69 views

Best random variable for infinite trials of a true/false event?

if you were to toss a fair coin a finite amount of times, and a success = heads, then the best random variable to represent it would be a binomial random variable. However, if you were to toss it an ...
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Computing SD for function of two RV's in R gives wrong results [duplicate]

I have 2 random variables, all independant, discrete and uniform I want to compute standard deviation for Z = X - Y Here is the R code: ...
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3answers
157 views

Decrease of $(X'X)^{-1}$ as n increases

Let $X$ be a $n \times p$ matrix ($n \geq p$ like a conventional data matrix), with each column j filled by iid draws from a variable $\mathcal{X}_j$. I would like ...
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53 views

Distance between random variables [closed]

I have found plenty of ways to compute the distance between random variables. However, I did not find any taking something else than the random variables as input. Do you know whether or not there is ...
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1answer
109 views

What is the domain of this random variable?

I've been self-studying Introduction to Statistical Learning. From page 16 of the book: "...suppose that we observe a quantitative response $Y$ and $p$ different predictors, $X_1$, $X_2$, $\ldots$,...
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3answers
99 views

Expectation of rational formula

I have two independent normal random variables $x$ & $y$ that are zero mean and unit variance. $a$ & $b$ are positive. I need to find the mean of $$z=\frac{ax^2y^2}{1 + bx^2}.$$ Any help ...
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37 views

Splitting up the variance of Z for Z = X*Y

$Z$ is a function of two dependent random variables, e.g. $X \cdot Y$. Here it is shown that $$var(Z) = var(XY)=(cov(X^2,Y^2)+E[X^2]E[Y^2])-(cov(X,Y)+E[X]E[Y])^2$$ I am interested in a metric that ...
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31 views

Modeling the joint distribution of stream statistics

I have a question regarding computing the joint discrete probability distribution of statistics in a number stream. I posted this problem in the Mathematics section as well but I'm hoping the ...
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0answers
31 views

Probability of X > Y and X > 0 (Frechet case)

I have two random variables, $X$ and $Y$. (They are both Frechet with different $\sigma$ parameter, but I'm interested in the general probabilistic argument here as well, so generality of answers is ...
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27 views

Converge in Probability of random variables

I don't know if I understood the (Convergence in probability of random variables) formula, why $ | Xn -X | $ should be $$ \geq \varepsilon $$ For example if $\varepsilon $=5( a random number), and ...
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195 views

Difference of Frechet variables

Let $$ X \sim Frechet(\alpha, s_1, m)\\ Y \sim Frechet(\alpha, s_2, m) $$ I'm trying to compute $Prob(X > Y$). This is equivalent of computing $Prob(X - Y > 0)$. Unfortunately, this is where ...
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35 views

Expectation of $\mathbb{E}(Tr(X^T A X))$ and $Var(Tr(X^T A X))$?

What is the expectation: $\mathbb{E}(Tr(X^T A X))$ and $Var(Tr(X^T A X))$ when $X_{i,j} \sim N(\mu, \sigma^2)$ and $X \in \mathbb{R}^{n \times k}$ where $n>k$ and $A$ is a given p.s.d matrix (not ...
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0answers
18 views

Sum of uniformly distributed random variables over different intervals?

Let $\{X_i\}_{i=1}^N$ be $N$ random variables uniformly distributed over the intervals $[a_i, b_i]$ respectively. How does the sum: $$\sum_{i=1}^N X_i$$ distribute? This is a generalization of the ...
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How to check if functions of i.i.d random variables are dependent or independent?

i'm new to this forum and the science of statistic.This is my question: Let's say that we have two i.i.d random variables X and Y, which both follow a Rayleigh distribution. Then, we define two new ...
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Lower-bound for total-variation between two sub-gaussian random variables

Problem setup Let $X$ and $Y$ random variables on the real line with following properties: $X$ and $Y$ are $\sigma^2$ sub-gaussian $E[X] = 0$ and $E[Y] = \Delta$. Question Whats the lower ...
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1answer
35 views

Check a sortition

[Warning: newbie in stats. It's a practical problem at work, not a homework, I'm not a student.] I have N entities and a process chooses M randomly (in theory...). M < N If there are two ...
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26 views

Finding conditional expected value

Given that X and Y are two independent exponentially distributed random variables with parameters a and b respectively. let Z = max(X,Y) find E[X|Z] attempt: I found that: P(Z=X) = b/(a+b) and P(...
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33 views

Can you explain why this distribution has been suggested for the length of a telephone call

I don't understand what the distribution in e) shows me? I would expect a normal curve about the mean but this is confusing
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1answer
25 views

Double Partial Derivatives of sum of variances of dependent random variables

I have the following function $$f(α)=Var[αX+(1−α)Y]=Var(αX)+Var[(1−α)Y]+2α(1−α)Cov(X,Y)$$ Partial derivative of this function w.r.t α leads us to following result $$f′(α)=2αVar(X)−2(1−α)Var(Y)+2Cov(...