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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158 views

Is the multiple of a Weibull distributed variable also Weibull distributed?

A Weibull distributed variable $X \sim \textrm{Weibull}(\lambda, k)$ has probability density function $f(x):$ \begin{equation} f(x;\lambda,k) = \begin{cases} \frac{k}{\lambda}\left(\frac{x}{\...
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13 views

random variable exponential [on hold]

Let the random variable X denote the point (distance) at which either of a motorcyclist’s tyres is destroyed. The probability that the motorcyclist can travel x miles without suffering non-repairable ...
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1answer
29 views

On finding a confidence interval. (exercise)

I am having some problems with the second point of this exercise Let $(X_1, \dots, X_n)$ be a random sample extracted from a population $X$ that is distributed as a uniform $(0, \theta)$ and let $Y_n ...
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2answers
28 views

Random Variables, Samples and Population

I am trying to understand the relationship of a Random Variable to a population and random samples. I understand that a Random Variable is a function that maps each event in a sample space to a number....
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0answers
58 views
+50

Calculating probability of an event involving two normally distributed random variables

I have two normally distributed, independent random variables $S^{(1)}$ and $S^{(2)}$: $$S^{(1)} \sim \mathcal{N}(\mu_1, \sigma_1^2),\ S^{(2)} \sim \mathcal{N}(\mu_2, \sigma_2^2)$$ Given two positive ...
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115 views

How can we get a normal distribution as $n \to \infty$ if the range of values of our random variable is bounded?

Let's say we have a random variable with a range of values bounded by $a$ and $b$, where $a$ is the minimum value and $b$ the maximum value. I was told that as $n \to \infty$, where $n$ is our sample ...
8
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2answers
784 views

Correlation between sine and cosine

Suppose $X$ is uniformly distributed on $[0, 2\pi]$. Let $Y = \sin X$ and $Z = \cos X$. Show that the correlation between $Y$ and $Z$ is zero. It seems I would need to know the standard deviation ...
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1answer
51 views

Finding pdf of transformed variable for uniform distribution

This is from MITx's Intro to Probability and Statistics course, the problem is on this page. Suppose $X \sim \textrm{Uniform}(0,1)$ and $Y=X^3$. Find the pdf for $Y$. Since it's a uniform ...
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2answers
78 views

Truthfulness of statements on the expected values of random variables

Are these statements true or false? Why? $E(|X|)\le 1 + E(X^2)$ $0≤|x|<1+x^2$ for all choices of $x$ with $x$ real number. What with $X$ random variable? if $E(X)<0$ and $ \theta \neq0$ ...
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9 views

Is it possible to do mediational analysis on trial-level data, including random subject and items factors?

I have words that are rated on several different dimensions. Each of say 30 subjects rates a set of say 30 words. I am curious if some feature of the words is related to some rated dimension of the ...
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18 views

Why use linear combinations of random variables?

I am in the process of learning bivariate distributions, especially bivariate normal disribution. I have noticed extensive use of linear combinations of random variables where the goal is to generate ...
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1answer
31 views

Can this situation really be modeled using a Binomial Random Variable? It seems we're sampling without replacement

I've seen the following situation modeled using a Binomial RV: We know that 20% of the people living in a village are against building an airport while 80% are in favor of it. If we randomly pick a ...
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1answer
142 views

Probability mass function and Random measure

I have seen this in a few tutorial and papers on Dirichlet Process, where people refer to the probability mass function of stick-breaking-process as a ...
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0answers
25 views

If a process is previsible, is the stopped process previsible?

Assume we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $X = \{X_n\}_{n \in \mathbb N}$ is an $\{\mathscr F_n\}_{n \in \mathbb N}$-...
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1answer
41 views

Prove divergence means that for some k, $\{|X_n - X| > \frac{1}{k}\}$

Probability with Martingales: What I tried: I think the hint is equivalent to $$\{\omega | X_n \rightarrow X\} = \bigcap_{k=1}^{\infty} \{\omega | \liminf_n [|X_n - X| \le \frac{1}{k}]\}$$
4
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1answer
68 views

Expectation of a function of a random variable from CDF

Is it possible to calculate the expectation of a function of a random variable with only the the r.v.'s CDF? Say I have a function $g(x)$ that has the property $\int_{-\infty}^{\infty}g(x)dx \leq \...
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1answer
27 views

Covariance between two random variables with different number of values?

Let's say I have a RV $X$ with values $100$, $200$ and their associated probabilities, and some RV $Y$ with values $35$, $47$ and $862$ with associated probabilities. What does it even mean to find ...
3
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1answer
46 views

Why are we using population symbols ($\sigma^2$, $\mu$, …) instead of sample symbols ($s^2$, $\bar{x}$, …) for Random Variables?

In my Stats course (can't ask prof, it's an online course), we got to random variables and the notation has changed from using $s^2$, $\bar{x}$, etc...to using what we were initially taught are the ...
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21 views

Changing the basis of random variables

Let $X_1$ and $X_2$ be two independent (1-dimensional) random variables and let $Y_1 = f^1(X_1, X_2)$ ($f^1$ is a deterministic function) be a (1-dimensional) random variable too. Question Does ...
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11 views

How to deal with “circular” or “modded” quantities like phases?

I have coefficients $A$ and $\phi$ for data of the form $\sin (A x + \phi)$. The $A$ is fine for all values, but the $\phi$ values I get are always in the interval $[0, 2\pi]$, i.e. they are always ...
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14 views

Method for stochastic data partitioning

I have one task. I have random polynomial like $F(x) = a_0(\omega) + a_1(\omega)x +\cdots+ a_n(\omega)x^n$, where $a_i(\omega)$ is a random variable for each $i = 1, \ldots, n$. Let looks for a ...
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1answer
51 views

generated two, specific random variables with a correlation

I would like to know which is the best method to do the following: - generate a random variable $Y$ over $Z_2$ (possibly following a Bernoulli distribution) such that $Y$ has correlation $\rho$ with a ...
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1answer
48 views

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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81 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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56 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
60 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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1answer
56 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 ...
3
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2answers
94 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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1answer
22 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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523 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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63 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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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
24 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
24 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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21 views

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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1answer
57 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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1answer
22 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
18 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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6 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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1answer
364 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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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
28 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
69 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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37 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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9 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 ...