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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Mixed effect modelling with multiple, nested random variable

Goal: comparing pitch (Hz) on three types of words Dependent variable: Hz Fixed predictor variable: word-type, points (measurements taken from five points on each token, to capture Hz change within ...
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27 views

product of independent random variables [duplicate]

This may be bit a little similar to the problem I asked yesterday, but I don't understand why I can't find my mistake. I am not familiar with the indicator function notation. Let $X$ be ...
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27 views

Linear transformation of a random variable by a tall rectangular matrix

Let's say we have a random vector $\vec{X} \in \mathbb{R}^n$, drawn from a distribution with probability density function $f_\vec{X}(\vec{x})$. If we linearly transform it by a full-rank $n \times n$ ...
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2answers
210 views

pdf of a product of two independent Uniform random variables

Let $X$ ~ $U(0,2)$ and $Y$ ~ $U(-10,10)$ be two independent random variables with the given distributions. What is the distribution of $V=XY$? I have tried convolution, knowing that $$h(v) = ...
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14 views

How to show via Delta method that the Linear Taylor series expansion of a normal random vector results in NORMAL DISTRIBUTION [duplicate]

How can it be proved using the delta method that the Linear Taylor series expansion of a normal random vector containing independent but NOT identically distributed elements results in a random ...
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28 views

If X is independent of Y given Z, is X/Z independent of Y?

When is the following true? If $X$ is independent of $Y$ given $Z$, $\frac{X}{Z}$ is independent of Y.
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1answer
45 views

How to find the distribution of a function of multiple, not necessarily independent, random variables?

If $Y$ is a random variable defined as $Y=g(X_1,X_2)$, where $X_1$ and $X_2$ are two different random variables whose distributions are known (say with pdf's $f_{X_1}$ and $f_{X_2}$), how do we find ...
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1answer
34 views

Generating random simulations of events [closed]

Whole-edited to make it more simple. Let's assume we have a concrete event: A baseball player's batting average is 0.32. I want to find a random number X, that ...
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123 views

Linear combination of discrete variables $T_i$ with $P(T_i=1)=P(T_i=-1)=1/2$

Let $T_1,...,T_n$ be iid with a Rademacher distribution; i.e., $P(T_i=1)=P(T_i=-1)=1/2$; and let $w = (w_1,...,w_n) \in \mathbb{R}^n$ without further constraints on $w$. Is there a way to compute ...
3
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3answers
165 views

The probability of a random variable being larger than a sequence of random values

Suppose we have a fixed, known, $n$, and each $x_1 \ldots x_n$ is a random number generated uniformly over $[0,1]$. What is the probability that $x_n$ is the largest value in the sequence?
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1answer
35 views

Can the population be smaller than the number of possible outcomes?

Say a random variable $X$ returns one of the values $1, 2, ..., 6$. Conceptually, does it make sense to speak of a population, where each individual element is just $1$? In other words, does the ...
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267 views

Why is this random variable both continuous and discrete?

The waiting time, $W$, of a traveler queuing at a taxi rank is distributed according to the cumulative distribution function, $G(w)$, defined by: $$G(w) = \begin{cases} 0 & \text{ for } ...
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73 views

tobit model please write down

Let $y$ be response variable taking on values $[0,1]$ , where $P(y=0)=0$ and $P(y=1)>0$ . An example is, for a random sample of markets indexed by i, $y_i$ is the share of the largest firm; by ...
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1answer
28 views

Expected value of the inverse of a random variable

Let $X$ be a random variable. $X$ can take the value 1 with probability $p$, and the value 2 with probability $1-p$. Can we write $E[\frac{1}{X}] = \frac{1}{E[X]}$? (note that $E[X] \neq 0$) Thank ...
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0answers
18 views

How to decide which of two distributions a sample is from?

I have two random distributions. How can I get a quantitative estimate of the likelyhood that a particular sample was taken from one or the other of the distributions? To be more precise: Let $I = ...
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1answer
39 views

Exclude Some samples for calculating CDF

I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and ...
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1answer
44 views

What is the expected number of coin flips, if you stop when the first coin flip is the same as the last?

In order to calculate the $\text{E}[X]$ where $X$ is the number of total coin flips, this is the approach I took: The probabilities are: $Pr(H) = p$ $Pr(T) = (1-p)$ Define indicator random ...
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2answers
59 views

Weakly correlated Random variables

If $N$ random variables are identically distributed but weakly correlated, in what condition we can approximate them as independent identically distributed (iid) ? I saw an old paper where based on ...
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1answer
19 views

How to compute $\mathbb{P}(A|B)$ of two independent RVs? [duplicate]

There are two independent RVs $X \sim \mathcal{U}(-1,5)$ and $Y \sim \mathcal U(-5,5)$. Let $A = \{ X \ge Y \land Y \ge -1 \land Y \le 1\}$ and $B = \{ X \le 1\}$. What is $\mathbb P (A | B)$? My ...
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1answer
77 views

Conditional expectation of $\mathbb{E}(X - Y | (X, Y)\in\mathcal{A})$

Given two independent random variables $X \sim \mathcal{U}[-1,5]$ and $Y \sim \mathcal{U}[-5,5]$, what is $$\mathbb{E}\{Y - X | X \le 1, Y > X, Y \in [-1,1] \}\,?$$ I managed to compute the ...
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1answer
32 views

Multiplication of two random distribution

I am trying to find the resulting PDF , when two random functions are multiplied. First function obeys normal distribution and second function obeys cauchy distribution. Can anybody tell me how to ...
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1answer
115 views

What is the correlation between X and X+Y?

If $X$ and $Y$ are two random variables, how do I calculate the correlation of $X$ and $X+Y$ in terms of $\rho$, $σ_x^2$ and $σ_y^2$ given that the $\text{Variance}(X)= σ_x^2$ and ...
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2answers
97 views

What is the meaning of the conditional $y|b$

I think I'm confused about a very simple thing. When we say that some variable is distributed as a Poisson distribution and we write $y \sim \text{Pois}(\lambda)$, is this the same that saying ...
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51 views

Example of a random variable that is not iid

What is an example of a random variable that is not i.i.d? The usual ones (coin flips, rolling of a dice) are all i.i.d, so I am trying to understand what is an example of a random variable that is ...
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1answer
126 views

(Co)variance of product of a random scalar and a random vector

Given a random scalar $ x \in \mathbb{R} $ and a random vector $ Y \in \mathbb{R}^n $ that are independent, can it be said that: $$ {\rm cov}(xY) = {\rm var}(x){\rm cov}(Y) + {\rm var}(x)E[Y]E[Y]^T + ...
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2answers
68 views

How to compute $\mathbb{P}(3x > -y > x \land x > 0 \land y < 0)$? [duplicate]

Knowing that both $x \sim \mathcal{N}(0,1)$ and $y \sim \mathcal{N}(0,1)$ ($x,y$ independent from each other), I want to compute $$\mathbb{P}(3x > -y > x \land x > 0 \land y < 0)$$ I'm ...
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1answer
71 views

pmf of random variable

There is a random variable that can take 3 values with the following probabilities: Pr(x=0) = 0.4 Pr(x=0.5) = 0.2 Pr(x=1)=0.4 How should i write the pmf of this ...
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1answer
50 views

Correlation under transformation

Suppose i have a random vector $X=(X_1,X_2,...,X_k)^T$ where each $X_i$ has cdf denoted by $F_i$ . The correlation matrix of this multivariate distribution is $R_k$. Define ...
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Uniform random variable as sum of two random variables

Taken from Grimmet and Stirzaker: Show that it cannot be the case that $U=X+Y$ where $U$ is uniformly distributed on [0,1] and $X$ and $Y$ are independent and identically distributed. You should not ...
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16 views

Relation of distributions

I want to predict a distribution using multiple related distributions. One method is to use multiple regression (the model specification is that the dependent variable, yi is a combination of the ...
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1answer
63 views

Do these random variables satisfy Lindeberg's condition?

I have the followig sequences: $Pr(X_n=n)=Pr(X_n=-n)=0.5$ $Pr(X_n=2^{n/2})=Pr(X_n=-2^{n/2})=0.5$ I have to show whether they satisfy Lindeberg's condition or not, but this condition is a bit ...
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25 views

How to simulate a random variable from a matrix variate distribution?

I am trying to simulate a random variable from the matrix variate normal distribution but have not seen any literature on it. If somebody could point me in the right direction for that or if they ...
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53 views

Transform two correlated random variable to independent variables without knowing correlation

I am thinking about this interesting question which arises in the following realistic setting. For example, in one medical experiment one drug and one placebo are applied to two randomized groups of ...
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24 views

Fast way to compute central moments of a Poisson random variable?

I am looking for a way to quickly compute the central moments of a Poisson random variable. I've found a couple of resources on how to compute the central moments, but I'm still trying to figure out ...
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17 views

Decomposition of a random vector into uncorrelated components

I have a set of random vectors $Y_i$ and their correlation matrix $C_{i,j}$. Each vector can be thought of as a sum of two uncorrelated vectors $Y_i=A_iX+B_iY$, where $X,Y$ are the same vectors for ...
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1answer
28 views

Properties of the minimum of several random variables

I've come across an interesting problem in my research that I don't quite know the answer to. Suppose I have a bunch of random variables: $$ X_1, X_2, X_3, ... X_N $$ They are not identical but they ...
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1answer
91 views

Random variables with some properties (conditional expectation)

I am looking for two random variables which fulfills the following two things: a) $\mathbb E(X|Y)<\infty$ and $\mathbb E(Y|X)<\infty$ b) $E(X|Y)> Y$ and $\mathbb E(Y|X)>X$ a.s Here is ...
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48 views

Using lmer for random effect and nested interpreting model comparison

I'm trying to do a lmer in r. I did an experiment in which I manipulated the temperature. My dataframe contains the following factors: Site: 11 level, random Species: 2 levels, fixed Subject: 10, ...
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1answer
28 views

MSE decomposition to Variance an Bias Square

In showing that MSE can be decomposed into variance plus the square of Bias, the proof in wikipedia has a step, highlighted in the picture. How does this work? How is the expectation pushed in to the ...
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2answers
56 views

Derive the LLN for a certain sequence

I have a sequence of dependent random variables $X_1, X_2...X_n$. Each RV is correlated with two other RVs and uncorrelated with the others.The ones that are correlated satisfy the condition ...
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37 views

What is the distribution of these functions of Nakagami random variables?

I am new to this forum and hope I can get help. A Nakagami random variable $X$ with parameter $m$ has the following pdf $$X\sim \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ ...
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1answer
14 views

Prevalence estimates based on randomized sample of clinical data

This is probably one of the more straight forward questions on here but here it is: I want to use a random number generator to sample X number of charts to look for the # occurrences of Y event. So ...
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asymptotic covariance between mean and standard deviation

I am trying to estimate the asymptotic covariance between mean and standard deviation. I know the following $$\sqrt n \hat \mu \xrightarrow{d}N\left( {\mu ,{\sigma ^2}} \right),\sqrt n \hat \sigma ...
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42 views

Example for uncorrelated, not independent , but with same distribution functions random variables

I am looking for an example of two random variables X and Y on $\Omega=\{-2,-1,0,1,2\}$ with the following properties: a) X and Y have the same distribution b) X and Y are uncorrelated c) X and Y ...
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78 views

Exponential random variable

The time it takes a printer to print a job is an exponential random variable with mean of 10 seconds. You send a job to the printer at 9:00 am, and it appears to be the fourth in line. What is the ...
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44 views

Sum of Bernoulli random variables

I need some help with a homework assignment. The question I'm given is: "Suppose that $X_1, X_2,..., X_n, W$ are independent random variables such that $X_i\sim Bin(1,0.4)$ and $P(W=i)=1/n$ for $i=1, ...
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1answer
136 views

How to bound a probability with Chernoff's inequality?

In my class, we were given Chernoff's inequality as $$P(X\le -t) \le e^{(-(\lambda t - \log( E(e^{-\lambda x}))))}$$ $$P(X\ge -t) \le e^{(-(\lambda t - \log( E(e^{\lambda x}))))}$$ It says that to ...
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82 views

Expectation of $(X + Y)^2$ where $X$ and $Y$ are independent Poisson random variables

I would really appreciate anyone's help with this problem: (let $E$ denote expectation) Suppose $X$ and $Y$ are independent Poisson random variables, each with mean $1$. Find: $E[(X + ...
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1answer
25 views

Sum of dependent R.V

I have two random variables whose PDF are parameterized by an unknown constant as follows: P(A;d) P(B;d) apparently, these two are not independent, so to find P(A+B;d) one cannot use convolution. ...
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46 views

sum of correlated random sample

Suppose I have 1000 draws each of two random variables X and Y. If I wanted to sample the sum of these variables, I would simply calculate 1000 samples, i.e. $$ S_{i}=X_{i}+Y_{i}, i=1,2,…,1000 $$ ...