A random variable or stochastic variable is a value that is subject to chance variation (i.e., randomness in a mathematical sense).

learn more… | top users | synonyms (1)

0
votes
1answer
26 views

Random variable variance

I have the model $y_i=\beta_1+\beta_2 X_i+ u_i$ where $u_i\sim\text{iid } N(0,\sigma^2)$. I estimate $\beta_1$ and $\beta_2$ by drawing a straight line between the first $(x_1,y_1)$ and last dot ...
0
votes
0answers
15 views

Mathematical notations for random variable and average from discrete set

I have a set data-set with the following structure, resembling ratings: UserID |ItemID | Score I have some algorithms that will predict the score of a user-item ...
4
votes
0answers
28 views

Curve-fitting and variance-stabilizing transforms

I'm working with data corrupted by a mixed Poisson-Gaussian noise model (for example with images gathered in astronomy or electron microscopy), and have been using the generalized Anscombe ...
2
votes
0answers
18 views

Estimating number of intersections between point and polygons

I have a 2D plane (a large rectangle) with a finite size in the x and y direction, which is the field of my problem. The field is covered by $n$ smaller rectangles that are located randomly within ...
4
votes
3answers
64 views

What is the correct notation if two random variables belong to the same distribution?

I want to explain that both the real and imaginary part of a complex variable follow a zero-mean complex Gaussian distribution. How can I write that? For one variable I'd write $a \sim ...
1
vote
0answers
38 views

probability that a variable is ONE OF the top k out of n when ordered

Suppose ($h_1,h_2,...,h_n$) is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a non-random variable which can vary across $i$ and $X_i$ is a random variable with Pareto Type I distribution. ...
0
votes
3answers
49 views

Biased coin toss simulation — which random generator is most appropriate?

I have a doubt whether the uniformly distributed random generator is the most appropriate to use for simulating biased coin toss. ...
3
votes
4answers
40 views

Conditional variance - $Var(X + U | X) = Var(U)$?

I am wondering if the following equality holds - $Var(X + U | X) = Var(U)$? where $X$ and $U$ are two independent random variables? It seems can we say $Var(X + U | X) = Var(X|X) + Var(U|X) = ...
3
votes
1answer
22 views

Notation: Deterministic Variable, Random Variable, Realization of Random Variable, Function

Is there a standard notation for distinguishing between the following: Random Variables Realizations of Random Variables Deterministic Variables (not random) Functions I am familiar with using ...
1
vote
0answers
25 views

How to apply the random statement in my model?

i am trying to model the following data in SAS using proc glimmix and i would like feedback if i have modelled my data correctly. My data consist of individual chickens (ID) who are grouped by ...
2
votes
2answers
35 views

Probability generating function for negative values of random variables?

What if we have negative integral values for a random variable?Then is it possible to write a probability generating function for it? All definitions I have seen so far is for non negative integer ...
1
vote
1answer
21 views

Laplace transform and density

It is true that the Laplace transform of a (positive) random variable characterises that random variable, just like its density? ($L_X(z) = E(exp(-Xz))$)
2
votes
0answers
29 views

Largest eigenvalue of this random 2x2 matrix

Consider four random complex vectors $\mu_i$ of length $K$ whose entries are drawn from the complex normal distribution $\mathcal{CN}(\mathbf{0},\mathbb{1})$ centered in zero and of unit variance. ...
0
votes
0answers
20 views

Measure of probability of obtaining a certain $\chi^2$ given the number of trials

I am calculating an equation with several random components a large number of times, and in order to select my final answer I choose that with the minimum value for $\chi^2$. I am hoping to find some ...
3
votes
1answer
94 views

Squared random variable $X^2$ vs $X\times X$

As I understand a random variable represents all possible outcomes of an experiment with their associated probabilities. Why $X^2$ is understood as squaring outcomes of experiments instead of as ...
2
votes
1answer
110 views

If a random variable V is independent of two independent random variables X and Y, how to prove that V is independent of X + Y?

This is question 3.8.4 of An Introduction to Mathematical Statistics and Its Applications, 5th Edition, by Larsen and Marx. This is not homework for a class I am taking now, but might someday be for ...
2
votes
1answer
42 views

correlation of two sums of random variables

Imagine two random variables $X$ and $Y$ which are correlated with $\rho = 1$. Both have a mean of $100$ and a standard deviation of $40$. Two other random variables $U$ and $V$ are correlated at ...
1
vote
1answer
25 views

Dirichlet sample by normalising Gamma RVs

I know that if you sample $K$ random variables $(X_1, X_2, \dots, X_K)$ from Gamma distributions using shape parameters $(\alpha_1, \alpha_2, \dots \alpha_K)$ and a scale parameter $\theta = 1$ such ...
2
votes
1answer
50 views

Rectifier function of random variable

Let $X$ be a random variable with distribution $F_X$ and density $f_X$. Define $$g(x) = \left\{ \begin{array}{lr} x & : x \ge 0\\ 0 & : x < 0 \end{array} \right\}$$ and let ...
1
vote
1answer
65 views

Probability function of three Random Variables multiplied, solidifying intuition

I ran across this exercise: Let $T$ be a random variable distributed as a $\text{Bernoulli}(p)$, $U$ be a random variable distributed as a $\text{Bernoulli}(q)$ and $W$ be a random variable ...
2
votes
1answer
59 views

Random variable as a function

I'm reading Schaum's outline of probability, random var. and random processes. In the second chapter they make it clear that a random variable $X$ is not a variable in the traditional sense but ...
3
votes
0answers
53 views

Markov Chain exercise in an exam

Suppose that $X_1, X_2, X_3... $ is a Markov chain with the following transition matrix: State: | 1 2 3 --+----------- 1 |0.2 0.4 0.4 2 |0.5 0.0 0.5 3 |0.6 0.3 0.1 (forgive my attempt at a ...
0
votes
0answers
14 views

Choosing levels of random factor in a mixed model

My apologies in advance if this question is as much about experimental design as it is about statistics. I'd like to know if there is a "best" way to choose levels of a random factor in a mixed model. ...
1
vote
1answer
53 views

Are explanatory variables considered random in PCA?

One of properties of PCA states that the sum of the variances of the principal components is equal to the sum of the variances of the explanatory variables. I wonder how to interpret this as I've ...
0
votes
0answers
9 views

Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$. Is it true that: If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...
1
vote
0answers
37 views

Random variables: True or False

I sometimes get confused when reading statistical definitions when they mention random variables (RV). Are they talking about a single draw? Are they talking about an estimator? To make things clear ...
0
votes
1answer
32 views

Decomposing a Combination of Distributions?

Let's say I'm studying the distribution/log-distribution of a random variable that may actually be the result of a combination of distributions. What are the ways I can check for this and possibly ...
1
vote
0answers
27 views

Average of several categorical random variables

Say I have a categorical random variable $X$ over a discrete label space $L$ = {Sky, Road, Tree, Unknown}. In other words each $X_i \in L$. Now I store the parameters of each $X_i$ which are the ...
0
votes
0answers
43 views

Calculating random effects from a glmerMod object (r package lme4)

Using the lme4 package, how does ranef() calculate (or extract) estimates of random effects from a ...
3
votes
1answer
47 views

Confidence interval for the “shift parameter” of a non-central exponential distribution

Suppose that $X_1, X_2, \ldots, X_n$ (with $n > 0$) is a random sample from a non-central exponential distribution with probability density function: $$f(x | \lambda, k) = \lambda * ...
1
vote
1answer
59 views

Maximum likelihood estimation for a sequence of observations

Assuming $A_1, A_2, \ldots, A_n$ are independent exponential random variables (each having the same parameter and, for the sake of simplicity, let's assume the value of the parameter is 1). Define ...
1
vote
1answer
68 views

Probability density function of the sample maximum of a random variable

According to my book, for a random sample $(X_1, \ldots, X_n)$ from a continuous distribution with p.d.f. $f(x)$ and c.d.f. $F(x)$, the p.d.f. of the maximum of the sample is $g(z)=nf(z)[F(z)]^{n-1}$, ...
0
votes
0answers
43 views

Quantifying variable importance for GLMM using hierarchical partitioning (in R)

I am interested in quantifying variable importance for a binomial logistic mixed-model regression. My model has 5 fixed effects, and 3 random effects (2 nested). I am doing model selection and ...
1
vote
1answer
30 views

How to decompose the probability of sum of a series random variables?

Let $A_1, A_2, \ldots$ be independent random variables with known distributions. I'm wondering if it's possible to equivalently decompose the probability above into separated constraints? For ...
2
votes
1answer
51 views

Difference of two independent gamma distribution

Given two independent random variables $X\sim\Gamma(s,r)$ and $Y\sim\Gamma(t,u)$, what is the distribution of the difference, i.e. $D=X−Y$? I assume that $s$ and $t$ are integers. How can I obtain the ...
5
votes
1answer
84 views

Probability distribution of functions of random variables?

I have a very stupid doubt: consider the real valued random variables $X$ and $Z$ both defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$. Let $Y:= g(X,Z)$, where $g(\cdot)$ is a ...
0
votes
0answers
22 views

A binomial random number generating algorithm that works when n*p is very small

I need to generate binomial random numbers: For example, consider binomial random numbers. A binomial random number is the number of heads in N tosses of a coin with probability p of a heads ...
0
votes
1answer
37 views

How do I prove the mutual independence of a set of random variables given another set of mutually independent random variables?

Suppose $X_{1},\,\ldots,\, X_{n}$ and $T_{1},\,\ldots,\, T_{r}$ and two sets of random variables, with each $X_{i}$ being categorical, and each $T_{j}$ being continuous taking on all values in the ...
9
votes
4answers
775 views

Can anyone clarify the concept of a “sum of random variables”

In my probability class the terms "sums of random variables" is constantly used. However, I'm stuck on what exactly that means? Are we talking about the sum of a bunch of realizations from a random ...
2
votes
2answers
60 views

Convergence in distribution results not deriving from central limit theorem?

In the stats class I took, all the results I have encountered about the convergence in distribution of some random variables are in one way or another consequences of the Central Limit Theorem. Out ...
0
votes
1answer
63 views

Random Variable

Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F) ...
7
votes
2answers
325 views

Functions of Independent Random Variables

Is the claim that functions of independent random variables are themselves independent, true? I have seen that result often used implicitly in some proofs, for example in the proof of independence ...
0
votes
2answers
82 views

Function of random variable

If I have a random variable $X$ which has mean $\mu$ and variance $\sigma^2$, what is the approximate expression of $log(X)$ and $\sqrt{X}$? Do I assume normal approximation or use Taylor expansion?
2
votes
1answer
54 views

showing a random variable has an exponential distribution

Let $X_{1},..,X_{n}$ be independent, each with a exp($\lambda$) distribution. Let $Z=min(X_{1},..X_n)$. Show that $n\lambda Z$ has an exp$(1)$ distribution. I calculate that $P(Z>z)=e^{-n\lambda ...
3
votes
2answers
203 views

Choosing a discrete non-uniform distribution for generating random integers

I have a list $l$ containing integers in the range $[1,max]$ On list $l$ I do an operation $isPresent(x)$ which return true if ...
3
votes
0answers
50 views

Deconvolution of sum results in negative numbers

Given $T=G+A$ where $A$ and $G$ are independent random variables, I'd like to estimate the distribution of $G$ given empirical (measured) distributions of $T$ and $A$. Of note: all three random ...
0
votes
1answer
28 views

tight bound of bernoulli sums with unknown dependency

Consider n random variables $X_1, \ldots, X_n$ all follow same bernoulli distribution of mean $p$. But the dependency of these variables are unknown (i.e., cannot assume that they are independent). ...
1
vote
1answer
71 views

Simulating Diffusion/Wiener Process with Random Walk [closed]

I hope this is the right section for this kind of questions. I am trying to simulate, with MATLAB, a diffusion model starting from a Random Walk. I am using a Random Walk with information increment ...
0
votes
0answers
19 views
0
votes
1answer
19 views

decomposition of independent random variables

If X and Y are independent random variables such that X = A + B and Y = C + D? Are the pairs (A, C), (B, C), (A, D), (B, D) also independent? By this I mean whether A and C are independent, B and C ...