Questions tagged [random-variable]
A random variable or stochastic variable is a value that is subject to chance variation (i.e., randomness in a mathematical sense).
2,372
questions
4
votes
1
answer
19
views
Sum of two i.i.d R.V having singly non-central F distribution
Noncentral F-distribution is used frequently in communication areas. In one of the applications, I need to do a sum of two i.i.d R.V having non-central F-distribution with parameter 1 (d.o.f for ...
- 41
2
votes
1
answer
25
views
Sampling from a distribution function $g_{x}$ that will follow $f_{x}$
I am using acceptance-rejection sampling to sample random variable $x$ according to distribution $f(x)$. The steps I followed are
First generated uniformly distributed random variable $x$ from 0 to $...
- 21
1
vote
0
answers
18
views
Including Random Slopes for Variables That Are Not Specified as Fixed Effects
I am interested in whether it is ever logical to specify a random slope of a variable that is not listed as a fixed effect in the model. For clarity, I have provided an example below.
10 Participants ...
- 43
0
votes
0
answers
18
views
Explain the intuition of Y = X/2 [duplicate]
I have some issues with understanding the intuition behind the following assignment.
Problem 2. When they say Y = X/2 what does this mean?
Does this mean whenever x takes any random value X, then y ...
- 163
1
vote
0
answers
22
views
Conditional probability given conditional probabilities [closed]
If $X$ and $Y$ are independent binomial random variables with identical parameters $n$ and $p$, show
analytically that the conditional probability of $X$, given that $X + Y = m$ is the hypergeometric ...
3
votes
1
answer
164
views
Is the variance of the mean of a set of independent random variables equal to the average of their respective variances?
I understand that, given a set of iid random variables, the variance of the sum is equal to the sum of the variance. Likewise, I know that the variance of the mean is equal to the variance over n.
My ...
1
vote
0
answers
54
views
Prove that a random variable follows MA(1) process
I have the follow variable
$$y_t = e_t + u_t + \theta u_{t-1}$$
Here $u_t$ and $e_t$ are mutually independent i.i.d
and $u_t \sim N(0, \sigma_u^2)$ and $e_t \sim N(0, \sigma_e^2)$.
I am trying to show ...
- 55
0
votes
0
answers
33
views
What is the variance of the new random variable created from taking the mean of one random variable minus another random variable?
I am having a debate with one of my co-workers about the variance of a new random variable created from taking the mean of one random variable minus another random variable.
The random variable R is ...
- 21
2
votes
1
answer
101
views
Existence of distribution that its difference of two iid RVs becomes a desired distribution
For any distribution, we can substract two random variables and find the distribution of the difference. But what about the reverse? Can the below statement be true in general or under any condition ...
- 43
8
votes
1
answer
128
views
Radial axis transformation in polar kernel density estimate
Consider a kernel density estimate of a continuous, non-negative random variable defined over the unit circle with no discontinuity between 360 and 0 degrees.
Unlike in the most common KDE ...
- 193
9
votes
3
answers
332
views
Density of $|t_1 - t_2|$ where $t_1$ and $t_2$ are iid with $P(t) = \alpha e^{-t\alpha}$
I am trying to answer the following question from my quantum mechanics textbook and my probability theory is admittedly rusty (this is not schoolwork as should be clear from my post history on Phys ...
- 203
7
votes
1
answer
581
views
Why must a product of symmetric random variables be symmetric?
I was reading about weight initialization in neural networks (He et. al, 2015) when I came across this statement:
"If we let $w_{l-1}$ have a symmetric distribution around zero and $b_{l} = 0$, ...
- 173
0
votes
0
answers
26
views
What does the quality of representation of a variable mean in PCA?
I understand that the quality of representation of an individual by a certain axis is measured by the cosine of the angle between the axis and the individual; the more the vector representing the ...
- 208
3
votes
2
answers
91
views
Case when random variable X and its square $X^2$ are independent
I have a random variable $X$, and I need to give an example when $X$ and $X^2$ are independent.
I can choose whatever distribution I please, I can say $X$ is a Bernoulli, uniform, or any other ...
- 389
1
vote
2
answers
45
views
Variance of the difference of two iid sample means
Let $X_{1}, ..., X_{n}$ be random variables independent of $Y_{1}, ..., Y_{n}$, where both groups are iid with associated population means $\mu_{1}$ and $\mu_{2}$ and population variances $\sigma_{1}^{...
- 13
1
vote
2
answers
30
views
Calculating the n-th moment of a RV, including negative fractional moments
I am stuck trying to solve the following exercise..
Let $X: \Omega\to [a,b] \subset \mathbb R$ be a uniformly distributed random variable. Compute the n-th moment of $X$, i.e. compute $\mathbb E[X^n]$ ...
- 11
0
votes
0
answers
25
views
Computing the expected value of a new Normal Random Variable (transformation)
I have the following exercise to do:
Let X be a normally distributed variable with mean μ and variance σ^2, i.e. X∼N(μ,σ^2). Define a new random variable to be Z=X^2−X. Compute the Expected Value of Z
...
- 11
0
votes
0
answers
35
views
Linear Combination of Bounded Pareto RVs
I am working with bounded Pareto distributions and was wondering whether I can say anything about the distributions of linear combinations of Pareto RVs? Suppose the PDF
$f(x; \alpha_i, L_i, H_i) = \...
- 55
4
votes
1
answer
230
views
random number generation of truncated multivariate normal distribution
I want to generate random numbers from truncated multivariate normal distribution specified as follows:
$ \begin{bmatrix}
Y \\
X
\end{bmatrix} \sim N
\begin{pmatrix}
\begin{bmatrix}
\mu_Y \\
\mu_X
\...
- 303
1
vote
2
answers
94
views
Dependency and correlation between a random variable and it's square [duplicate]
I have the following question. At first this seemed very silly but after thinking about it, I found my self struggling.
Given $X$, a random variable, I should decide if the following sentences are ...
- 413
0
votes
1
answer
22
views
What does the parameter s in Sum-of-Gamma mean?
So the sum of gamma, introduced in I-MLE is defined as the following:
$$SoG(k,t,s)=\frac{t}{k} \left( \sum^{s}_{i=1} Gamma(1/k,k/i) - \log(s) \right)$$
But what exactly is $s$? It clearly controls the ...
- 1,108
0
votes
0
answers
36
views
What is the distribution of sum of complex Normal R.Vs which are independent?
This may be a trivial question. But I am fairly new to statistics and distributions. I am trying to find analytical solution to sum of complex independendent Gaussian R.Vs following Rician ...
- 21
0
votes
1
answer
28
views
How to compute the variance of the following expression?
I have a complicated function that looks like this.
$$ L = \sum_{m = 1}^{M} \left|\frac{\sin\left(N\frac{x_m}{2}\right)}{\sin\left(\frac{x_m}{2}\right)}\right|^2 + \sum_{p = 1, p \neq q}^{M}\sum_{q = ...
- 71
2
votes
2
answers
52
views
Finding a non-symmetric test for correlated random variables $A$ and $B$ where $B\ge A$
Let's say I randomly sample pairs of numbers, $(A, B)$, each within a range of 0 to 10. In every pair, $B \ge A$. If I were to graph these points, they would fill only the half of the square that lies ...
- 21
0
votes
0
answers
48
views
Doubt in empirical distribution, sample and random variable
I have a basic doubt in sample and random variables. I have read related posts on this site but still some doubt is still left.
Suppose we have a population and we are drawing some entries from it ...
- 115
4
votes
5
answers
316
views
Why are $X \sim U(-1,1)$ and $Y=X^2$ dependent?
Suppose we have two continuous random variables $X \sim U(-1,1)$ and $Y=X^2$. I don't understand why they are dependent.
$$E[X] = 0$$
$$E[Y] = \int_{-1}^{1} x^2 dx = 2/3$$
$$E[XY] = \int_{-1}^{1} x^3 ...
- 143
4
votes
2
answers
308
views
correlation: the difference of two correlations is positive, but the correlation of the difference is negative?
There are 3 random variables, $X_1$, $X_2$ and $Y$. We know $$corr(X_2, Y)>corr(X_1, Y)>0$$, but $$corr(X_2 - X_1, Y)<0$$
In other words, $X_2$ is more positively correlated to $Y$ than $X_1$,...
- 43
1
vote
0
answers
35
views
Log-concavity of a discrete random distribution
I'm working with a discrete distribution on the set of non-negative integers. The step-wise cumulative distribution function at any non-negative integer $i$ is
$$
F_i = \frac{2}{\log w}[\log(1+w^{i+1})...
- 31
9
votes
2
answers
812
views
How would you write mathematically that a random variable follows some unknown distribution?
For example, if I had a random variable X and knew it followed a normal distribution, I might write:
$X \sim N(5, 10)$
What would I write if I'm trying to say that the distribution is unknown?
$X \in \...
- 545
1
vote
1
answer
62
views
What is the probability the expected value is undefined or infinite?
What is the probability from a uniform probability measure (pg.37) on sample space $\left\{N(\theta,1)|\theta\in[0,1]\right\}$ that for some random variable $X$ in the sample space, the Expected-Value ...
- 161
0
votes
0
answers
34
views
How to interpret the loss function $\mathcal{L}$ as i.i.d. random variables?
$\mathcal{D}$ is the fixed but unknown distribution of the data. Usually, this refers
to the joint distribution of the input and the label,
$$
\begin{aligned}
\mathcal{D} &= \mathbb{P}(\...
- 715
15
votes
4
answers
1k
views
Is the median preserved for any strictly monotonic mapping?
Problem
Let $X\sim f_X$, where $f_X$ is the probability density function of $X$. Let $g: \mathbb{R} \to \mathbb{R}$ a strictly monotonic (decreasing or increasing) mapping. I aim to prove or disprove:
...
- 1,351
2
votes
0
answers
27
views
How to empirically check a PDF
Intro
Let $Y$ be a random variable whose PDF is $p_Y(\cdot)$. Let's say that $Y$ is a function $g(\cdot)$ of another random variable $X$ whose PDF $p_X(\cdot)$ is given. Then, you do your calculation ...
- 131
0
votes
0
answers
12
views
Nested or Crossed random effects?
I am conducting an experiment in which participants come twice to the lab to perfom a task. So we have two different experimental sessions, but participants do the same thing in both sessions. The ...
- 1
1
vote
1
answer
51
views
How to combine two integrals containing the PDFs of a variable and its linear transform?
Original Post:
Suppose we have two random variables $X$ and $Y$ with cumulative distribution functions $F(x)$ and $G(y)$. We know that $Y = aX + b$.
I want to compute $Z(x) = F(x) - G(y)$.
What I have ...
- 13
3
votes
1
answer
44
views
If X and Y are independent and $E[(XY)^2] = 0$, then $P(X = 0) = 1$ or $P(Y = 0) = 1$
Let $(X, Y)$ be a discrete random vector. Prove that: If $X$ and $Y$ are independent and $E[(XY)^2] = 0$, then $P(X = 0) = 1$ or $P(Y = 0) = 1$
Since $X$ and $Y$ are independent, covariance, i.e., $E(...
- 207
3
votes
3
answers
94
views
Variance estimation for small sample size
The following variance estimator of a set of data points $x = (x_1, ..., x_N)$
$$
\text{Var}\,(x) = \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2
$$
has itself a large variance when $N$ is small (in my ...
- 31
1
vote
1
answer
29
views
How to handle physically meaningless values in sampling?
I'm working on stochastic optimization for optimal energy dispatch, where the uncertainty of photovoltaic power output should be considered with monte carlo sampling and scenario reduction technique. ...
- 113
0
votes
2
answers
40
views
Why the calculation of a probability distribution mean works
We know that the mean of a random variable is calculated by adding up the multiplication of the random variable values with their related (probability distribution) probabilities.
But what I am trying ...
1
vote
0
answers
13
views
Estimating Joint Probability from limited data
This is probably a broad question but I am interested in getting the Joint PDF for n random variables each of which can take integer values within a finite range.
I have data in the form of time ...
3
votes
1
answer
102
views
Correlation vs Conditional Expectation
I have two random variables, $X$, and $Y$. What are the differences or similarities between Corr$(X, Y)$ and $\mathbb{E}(X|Y=y)$. Can I deduce one from the other?
- 33
1
vote
2
answers
104
views
How to identify variables of a dataset?
i have a dataset of the annual sales of a bakery and i'm confused about how many variables it has. is it the number of products or the number of years? can someone help me out? thank you!
user376745
4
votes
1
answer
132
views
How to Find PDF of Transformed Random Variables Numerically?
Is there a way to compute and plot the transformed random variable in Python or R?
Let's say if $X_1$ and $X_2$ are independent and identically distributed random variable with PDF defined over non-...
- 43
2
votes
1
answer
61
views
Is Z a random variable of X?
Is Z a random variable of X if it is transformed from X? If Z is a random variable, and therefore a function, of X, could one denoted Z as Zₓ(x)?
1
vote
0
answers
35
views
Probability of a "successful" arrival over time from $0$ to $t$?
I have this problem that I have been trying to solve using a random process (such is the task).
There are $n$ keys. Only one of them opens the lock. The keys are tried without replacement. Find the ...
- 11
1
vote
1
answer
93
views
Equivalent of $E[(a-X)^2] = E[(a-E(X))^2] + Var(X)$ for $E[|a-X|]$ and $med(X)$?
The minimzer of the MSE $E[(a-X)^2]$ is $a=E(X)$, and the MSE can be decomposed into $E[(a-X)^2] = E[(a-E(X))^2] + Var(X)$.
I am wondering whether there exists a similar expression th MAE $E[|a-X|]$ ...
- 385
1
vote
0
answers
47
views
What is the domain of the estimates of the parameters in linear regression?
An answer here stated the following:
Here is another place where typical regression teaching sources are slippery. In some cases, they refer to the estimates $\hat \beta_0$ and $\hat \beta_1$ as ...
- 208
1
vote
2
answers
58
views
Does the average of a random sample minimizes MSE when you "know nothing about the distribution"?
Consider any random variable $X$ and any random sample $(X_1,\dots, X_n)$ such that $X_i \sim X$.
As is well-known, $E(X)$ is the constant that minimizes the MSE of $X$, i.e., $E(X) = \arg\min_a E[(a-...
- 385
4
votes
2
answers
220
views
Operations on Random Variables vs Distributions vs Random Samples
What is the difference between i) random variables, ii) distributions of random variables, and iii) random samples?
While trying to figure out how to average random samples from various random ...
- 1,791
1
vote
1
answer
103
views
What is the correct way to average random variables and get correct quantiles
Say I have two random variables $A$ and $B$ which may or may not be independent. I also have their $0.95$ quantiles $Q95_A$ and $Q95_B$. What is a valid way to average these densities and obtain valid ...
- 1,791