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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Derive the conditional probability $P\left(X \in A \middle| \, Y \in B \right)$

I am trying to derive the following form of conditional probability: $P\left(X \in A \middle| \, Y \in B \right)$. Let $g$ be a complex r.v. normally distributed as $\mathcal{CN}(0,1-\sigma^2)$, ...
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1answer
20 views

What are the recent real life use or applications of the Cauchy Random Variable?

We have a short assignment on the described question and I already have gone through a lot of trash results from Google. I can't seem to find any. I don't know where else to post this question. ...
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2answers
52 views

Distribution of $e$ if $g=\tilde{g}+e$, $g=|f|$, $f \sim \mathcal{N}(0,1)$ and $\tilde{g}$ is the quantization of $g$

Let $f \sim \mathcal{N}(0,1)$ be a normal random variable with zero mean and unit variance. Let $g=|f|$. Let $\tilde{g}$ be the quantization of $g$. We suppose that there are $n$ possible levels of ...
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15 views

Need help establishing a baseline

I'm currently working on a project that involves the computation of a (random) baseline. I want to estimate the mean & variance of a function of a random vector. The function involved doesn't ...
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18 views

Bound on the variance of a product

Let $Z$ be a positive $\mathbb R$-valued random variable bounded above by $M>0$, and $H$ an $\mathbb R^d$-valued random variable (seen as a column vector) such that $\mathbb E[H_i^2]=1$. Define the ...
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13 views

Notational issues for point estimates

In the most basic form, (as I recall), consider a random variable $X$ which is defined over a probability space $\Omega$. Now, let us call a realization of $X$ as $x$ . As such, we can define ...
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14 views

Decorrelation of variables without Cholesky

To correlate variable together, for example to achieve a desired covariance, you can use either Cholesky decomposition or eigenvectors and eigenvalues. To decorrelate correlated variables, (in my ...
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115 views

How to generate two groups of $n$ random numbers in $U(0,1)$ such that sum of these two groups equal?

I want to have two groups of $n$ random numbers $u_i$ and $v_i$ in $U(0,1)$, such that $\sum u_i = \sum v_i$ What I tried is: I can firstly get $u_i$ by $U\sim U(0,1)$, make $s=\sum u_i$. Then I ...
2
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1answer
39 views

Random sampling from a population with mutually dependent elements

After reading this answer, I found myself confused. Say I am looking for the percentage of people who are likely to vote for a particular candidate. I am sampling from a population which is known to ...
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14 views

Unknown Variable in Neural Network

I was reading a paper from 1996 (http://www2.cs.uregina.ca/~jtyao/Papers/marketing_jisi.pdf) where an Unknown variable was used in the ANN that apparently caught information and influences not ...
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16 views

sum of dependent random variables [closed]

I want to find a function that determine connection between x and y that x and y both of them are determine by a common other factor. in fact x and y are dependent variables. function is like a ...
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22 views

Time series and images : difference and terminology

A time series is an ordered collection of random variables. Considering a one-dimensional time series $A_i = {a_{i1},a_{i2},\ldots,a_{it}}$ where $t$ denotes the time index. So, the time series is a ...
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0answers
18 views

Absolute error or RMSE when you know the exact value?

I'm testing the $k$ neighbour correlation of a uniform random sequence $x_i$ in $[0,1)$. I know its exact expected value to be $1/4$ and I want to show that the error decreases with $\sqrt{N}$ where ...
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2answers
34 views

Conditional probability of a single random variable

I have a Gaussian random variable $X$ and I have been told to find $P[x < a | x > b]$ and $P[|x|>c]$. What does $P[x < a | x > b]$ mean in terms of a single variable? Is ...
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45 views

Write $\mathop {\mathbb E}[(X AX^h)]$ in function of $\mathop {\mathbb E}[(X X^h)]$?

Suppose $X$ is an $i \times j$ random matrix. In addition, $X$ has complex i.i.d. normal entries with $0$ as mean. We define $A$ (of dimension $j \times j$) as a deterministic matrix. Is it ...
3
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57 views

Prove $\sigma^2(aX+b)=a^2\sigma^2(X)$ for discrete random variable

I'm a total newbie to statistics/math in general, so please bare with me. I'm currently reading the book The Cartoon Guide to Statistics, and stumbled upon the following statement (p. 69): $$ ...
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1answer
106 views

Distribution of $|q|^2$ if $\Re[q]\sim \mathcal{N}(\mu_1,\sigma^2/2)$ and $\Im[q]\sim \mathcal{N}(\mu_2,\sigma^2/2)$

Let $q$ be a complex random variable such that: $\Re[q]\sim \mathcal{N}(\mu_1,\sigma^2/2)$ and $\Im[q]\sim \mathcal{N}(\mu_2,\sigma^2/2)$. What is the PDF and CDF of the squared norm $|q|^2$ ?
3
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1answer
34 views

What do the realizations of X(t)=Usin(t)+Vcos(t) where U and V are random variables with mean 0 and and variance 1 look like from -2pi to 2pi?

I'm not sure what the realizations of a time series really mean, and how to implement any kind of drawing with random variables. Any hints or descriptions would be very helpful.
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0answers
24 views

Convergence of a sequence of Binomial variable with changing probability

Consider a $t\in(0,1)$. Consider, for $\Delta>0$ the random variable $X_t^{(\Delta)}$ defined as $$ \mathbb{P}[X_t^{(\Delta)}=1]=\left(1-\lambda\,\Delta\right)^{\left\lfloor ...
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1answer
33 views

Account for pedigree effects in linear mixed model

I'm a student in biomedical engineering. I wanna analyze difference of brain size between controls and patient. But it is very hard to me because there are some siblings in data. I would like to ...
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0answers
9 views

Generate random variables with predefined correlation structure AND fixing some values (followup)

This question is basically a followup to this one: Generate random variables with predefined correlation structure AND fixing some values because I can't follow the answer given there. I would like ...
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1answer
33 views

Random unit vectors in $\mathbb{C}^n$ and $\mathbb{R}^{2n}$

Suppose $\mathbf{u}\in\mathbb{C}^n$ is a complex random vector with circular symmetry, uniformly distributed on the unit complex $n$-sphere, so we have $\|\mathbf{u}\|=1$. In other words, $\mathbf{u}$ ...
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50 views

Shall I use a random effect or not?

I need to see if in the case I am going to present it is worth to use a random effect or not. I carried out some bird counts from 9 elevated lookouts in an island. Just to orient you, these lookouts ...
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1answer
31 views

Is $E[1_A | \mathscr{F_t}] = 0 ~\text{or} ~ 1 \ \Rightarrow E[1_A | \mathscr{F_{s}}] = E[1_A | \mathscr{F_t}]$ is only almost surely?

Spin-off from my previous question: Prove/Disprove $E[1_A | \mathscr{F_t}] = 0 ~\text{or} ~ 1 \ \Rightarrow E[1_A | \mathscr{F_{s}}] = E[1_A | \mathscr{F_t}]$ Apparently the conclusion holds true ...
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2answers
67 views

Why does a probability of 0 or 1 remain unchanged with new information, intuitively?

Related to these questions: Prove/Disprove $E[1_A | \mathscr{F_t}] = 0 ~\text{or} ~ 1 \ \Rightarrow E[1_A | \mathscr{F_{s}}] = E[1_A | \mathscr{F_t}]$ Does an unconditional probability of 1 or 0 ...
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0answers
59 views

Random permutations [closed]

Let $a_1 < a_2 < · · · < a_n$ denote a set of n numbers, and consider any permutation of these numbers. We say that there is an inversion of $a_i$ and $a_j$ in the permutation if i < j ...
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18 views

Confusion about difference between policy gradients of trajectories under stochastic/deterministic policies

(If this isn't the right SE I'm sorry! Please point me in the correct direction) I'm reading this survey on policy search in robotics, but I'm having a tough time with one part. I'm going to assume ...
4
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2answers
93 views

Why is “the probability that a continuous random variable equals some value always zero”? [duplicate]

I found lots of references that say, "the probability that a continuous random variable equals some single value is always zero". Why is that? Here is a counterexample I thought of: supposing $X\sim ...
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17 views

Sum then Division of iid random variables

How do I combine the distributions to add and divide this: ($x_1$ - $x_2$)/($x_1$ + $x_2$) - what, if any, effect would it have if I squared the numerator? Both are, separately, iidand ~N(0,1). ...
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1answer
27 views

Random Variable Transformation

Consider a Probability density function f(x) which is uniformly distributed between say (-5,5).Now if i define a random variable "y" which is related to the random variable "x" as follows: y = 1 ; ...
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1answer
51 views

Find P(Y=y | X=x) when X is a continuous random variable

could someone help me understand how to find the probability $\Pr(F=f_1 | X=x)$ by using the inputs below, where $X$ is a continuous random variable? Note: I know that probabilities of specific ...
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51 views

Finding what happened between two PDFs from the parameters of a resulting PDF

Assuming $X$ and $Y$ are Gaussian random variables with PDFs of $f(X)$ and $g(Y)$ with parameters of $(\mu_x, \Sigma_x)$ and $(\mu_y, \Sigma_y)$, we know that: for the operation of sum ($+$), if ...
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0answers
12 views

Notation of marginal distribution for random vector variables

I am reading this article about multilabel classification. Unfortunately I am unable to understand the following notation, as found in section 2. Multilabel Classification: $P^{(i)}_x (Y_i) = ...
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1answer
46 views

Probability of absolute value of a sum of two symmetric random variables

Suppose that X and Y are independent and identically distributed random variables with probability density function f(x)f(x) that is symmetric about the origin. we have P[|X+Y|≤K]>=a. can I show that ...
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37 views

Find $\mathbb{E}\{M W V V^H W^H M^H \}$ in the following case

Let $M$ an $n \times p$ matrix with complex Gaussian elements with mean $= \mu$ and variance $= \sigma^2$. Also le define matrix $W$ as an $p \times m$ matrix for which the columns are of unit norm. ...
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1answer
26 views

Bayesian approach and ML approach for distributions

Suppose that we have two sets of discrete random variables X ~ f(θ), Y~g(θ) where X and Y are independent, and the parameter θ is the same in both cases. We are interested in predicting Y on the basis ...
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35 views

Partial Derivatives of Function of two Random Variables with respect to the Random Variables

We have a function $G(X,Y,\theta)$ of two random variables, $X$, $Y$ and parameter $\theta$. Probability Density function of $X$ and $Y$ is given by $f_{X}$ and $f_{Y}$ respectively. Cummulative ...
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2answers
83 views

Describing random variables: “Defined as” contrasted to “has the property of”

This was inspired by this question and the comment of user @Did to it. At first it may appear as some subtlety that would interest some educational course only. But since it relates to how we ...
2
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0answers
55 views

What is the value of E(S)?

Let $x_1,x_2,..x_{21}$ be a random sample from a distribution having variance 5. Let $$\bar X = \frac1{21}\sum_{i=1}^{21}X_i\quad\text{and}\quad S=\frac1{21}\sum_{i=1}^{21}(X_i-\bar X)^2$$ What ...
5
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2answers
83 views

intuition behind exchangeability property, and its use in statistical inference

I'm reading "Bayesian Data Analysis" by Gelman et al., and I encountered this exchangeability property: $\{X_n\}_{n \in N}$ is exchangeable if $F_{X_1,\ldots,X_n}(x_1,\ldots,x_n)$ is symmetric in its ...
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1answer
23 views

Generating random variables with inverse df

Assume a distribution $X$, and we know the idf $F^{-1}$ of $X$. Let $U \sim U(0,1)$. Why is drawing an element from X according to $F^{-1}(u) = Z$ considered to be more random than just drawing an ...
5
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2answers
62 views

What is the moment of a joint random variable?

Simple question, yet surprisingly difficult to find an answer online. I know that for a RV $X$, we define the kth moment as $$\int X^k \ d P = \int x^k f(x) \ dx$$ where the equality follows if $p = ...
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1answer
59 views

let x and Y be independent standard normal random variables . then what is the distribution of U= [(x+y)/(x-y)]^2 [closed]

let x and Y be independent standard normal random variables . then the distribution of U= [(x+y)/(x-y)]^2
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1answer
24 views

Test for RVs with known probabilities?

I have written code that generates a sequence of distinct integers. The integers are assumed to occur in the sequence with fixed probabilities. For example, if the sequence contains the numbers ...
2
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0answers
39 views

Derive the distribution of the fraction $\frac{[M^+ (M^+)^h]_{i}}{ || M^+||^2}$

Let $M^+$ represents the pseudo-inverse of matrix $M$; $M^+=(M^hM)^{-1}M^h$, where $h$ denotes the conjugate transpose. We assume that the elements of $M$ are complex Gaussian with zero mean and unit ...
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1answer
45 views

Estimate $m$ using method of maximum likelihood

Estimate $m$ using method of maximum likelihood. In the box there are $91$ balls, where $m$ are red, and the rest are blue. To estimate unknown parameter $m$, at once $19$ balls are drawn, $7$ being ...
2
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2answers
43 views

Quasi and True Experiments

If subjects are recruited using a convenience sample to take part in a study, and each participant is then randomly assigned to either treatment condition, A and B. (i) Will my result be ...
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0answers
19 views

How to find the variance of a discontinuous function?

Let's say I have a standard normally distributed random variable X. Let's say I have a function f(X) such as the following: ...
2
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0answers
24 views

How can I estimate the statistical error of a sum?

I'm summing loan application amounts for a fixed time frame. Is there a way to attach an error on the sum? I was thinking of assuming some distribution for the amounts and estimating the population ...
12
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2answers
307 views

Is Slutsky's theorem still valid when two sequences both converge to a non-degenerate random variable?

I am confused about some details about Slutsky's theorem: Let $\{X_n\}$, $\{Y_n\}$ be two sequences of scalar/vector/matrix random elements. If $X_n$ converges in distribution to a random ...