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).
1,899
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7 views
Condition for zero or non-zero bias under transformation of random variable?
Suppose $\epsilon$ is a zero-mean univariate random variable noise with finite variance. I would like to find the condition on the function $f$ so that:
$$ \exists x \in R \ \mbox{s.t:} \mathbb{E}[f(...
2
votes
1answer
42 views
Product of a Gaussian by a Beta random variable
I'm trying to find the distribution of a random variable $Z = X \cdot Y$, where $X \sim N(\mu,\sigma^2)$ and $Y \sim \text{Beta}(\alpha,\beta)$ with $\alpha$=1.
I have tried with the convolution ...
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0answers
21 views
Linear mixed effect model: Small and unbalanced number of repeated measures per individual
I have a data set which contains 2 or 3 repeated measurements taken from 50 individuals over a 6 year period. The time between measurements is inconsistent.
My goal is to estimate the effect age (...
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12 views
Population distributions/data generating functions in Bayesian Statistics
In many frequentist stats courses, random variables come from some distribution at the population level and as such we could say that $y=X \beta + \epsilon$ is the true function for something like ...
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0answers
22 views
How can $X$ be a discrete random variable? [duplicate]
Suppose that the cumulative distribution function of discrete random variable $X$ is given by,
$$F(x) =
\begin{cases}
0 & \text{$x$ < 0 } \\[1.5ex]
\dfrac{x}{4} & \text{$0 \leq x<1$}\\[...
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1answer
68 views
Mann-Whitney Normal Approximation process help
Let $X_{1}, X_{2}, ..., X_{n}$ is i.i.d sample from $X$ and $Y_{1}, Y_{2}, ..., Y_{m}$ is i.i.d sample from $Y$. And both samples are independent each other.
Trying Mann-Whitney U-test then, $U =$ $\...
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7 views
Derivation of Pearson's Product Moment Correlation Coefficient's (PPMCC) distribution from bivariate normal variables?
I'm interested in reading through the derivation of the probability density function that PPMCC follows when it's input is a bivariate normal variables. Mathworld gives the following equalities:
$$P(r)...
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2answers
43 views
Is this problem calculable only due to the parameter choices?
I am looking at a problem form Hogg, Tannis & Zimmerman (Ed. 10), and I am curious if the given problem is calculable (for an upper-level undergrad math/stats course) because of the choice of the ...
2
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1answer
48 views
Show $X_n \to 0$ in probability
I am asked to show :
Let $X$ be a real-valued random variable on $(\Omega, F , P)$ and define
$X_n(\omega) = nX(\omega)$ if $n<X(\omega)\le n+1$ and $0$ if else. Prove that $X_n \to 0$ in ...
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1answer
30 views
Monte Carlo simulation for grouped averages [duplicate]
Assume we have $N$ random variables $X_1, \ldots, X_N$. As an example, assume that these random variables describe test scores of $N$ students. I am interested in finding the distribution of average ...
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1answer
30 views
What does $X_1,…,X_n $ mean in $X_1,…,X_n \sim N_p(0,\Sigma)$ (iid)?
What exactly does $X_1,...,X_n$ mean in $X_1,...,X_n \sim N_p(0,\Sigma)$ (iid) ?
I am confused, since what I imagine is that the variables $X_1,...,X_n$ are the columns of a dataset? But From the fact ...
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0answers
27 views
If X and Y ~ Uniform(0,1), what is the distribution X/Y? [duplicate]
Given that $X$ and $Y$ are random variables drawn from a uniform distribution between $0$ to $1$, i.e.
$$ X \sim Uniform(0, 1) \\
Y \sim Uniform(0, 1)$$
And given that $Y \neq 0$, what do we know ...
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1answer
44 views
Show That $\sum_{K=1}^{n}\frac{X_k}{n^{\frac{1}{\alpha }}}$ If ${X_n}$ is $X_k$s Same Distribution
Let ${\{X_n}\}$ be a sequence of independent random variables and the
stable distribution of order alpha $(0\le\alpha\le2)$.
Show that $$\sum_{k=1}^{n}\frac{X_k}{n^{\frac{1}{\alpha }}}$$ if
${X_n}$ is ...
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4answers
94 views
Show That $Cov(X,\frac{1}{X})\le0$ if $X$ Is Positive Random Variable?
I Know That , If $X$ and $Y$ are independent then $Cov(X,Y)=0$ And $ Cov(X,Y)=E(XY)−µ_Xµ_Y .$
But I Don't Know How to prove The $<$ part And How to prove the part for positive random variable.
Can ...
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1answer
898 views
Why the sum of two absolutely-continuous random variables isn't necessarily absolutely continuous?
Why "a sum of two absolutely-continuous random variables does not need to be absolutely continuous"?
See problem 6.4 on page 6 in https://web.ma.utexas.edu/users/gordanz/notes/...
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0answers
16 views
Need to look at ordered samples for computing variance of sample mean in simple random sampling without replacement
In simple random sampling without replacement, the space of all possible unordered samples has cardinality $N \choose n$ and the space of all ordered samples has cardinality ${N \choose n} n!$. When ...
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2answers
50 views
Showing that $P(X_1>X_2) = \int_{0}^1 P(X_1>X_2 | X_2=x) f_{X_2}(x) dx$
I am going through this post in trying to prove the probabilistic interpretation of the AUC for a ROC Curve (for a classifier):
The AUC for a ROC curve is the the probability of the classifier ...
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1answer
31 views
Binomial distribution vs. Discrete uniform distribution. vs. randomly selecting
I have an array of $n$ non-negative integers where each element can be randomly (uniformly) selected with repetitions from all the integers between 0 and $m$ (with $m>n$). So the probability of ...
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0answers
106 views
On random variables made up of independent random digits
Some random variables can be expressed as a binary expansion whose digits are chosen independently at random; this is called a convolution. One example of this kind of random variable is the one for ...
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0answers
41 views
Testing uniform distribution of a discrete function derived from a random number generator
(Forewarning: I'm a programmer, not a statistician, so I apologize in advance for any misuse of terminology!)
I'm testing a known random number generator that implements the PCG algorithm. This RNG ...
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1answer
39 views
Can one uniformly generate complex numbers of absolute value less than a given constant $R \neq 1$? [duplicate]
Can one uniformly generate complex numbers of absolute value less than a given constant R?
This would appear to be equivalent to picking points $(x,y)$ uniformly in a disk of radius R, where $x$ is ...
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0answers
29 views
Uncertainty principle in probability theory
In probability theory, there is the covariance inequality
$$\operatorname{Var}(Y) \geq \frac{\operatorname{Cov}(Y,X)^{2}}{\operatorname {Var} (X)}.$$
In signal processing, there is a similar ...
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0answers
11 views
Technical name for difference between largest and second largest element in sequence
Let $X_1,\ldots,X_n$ be a sequence of $n$ real numbers (for example, iid random variables, etc.) and let $X_{(k)}$ be the value of the $k$th largest number, so that $X_{(1)} \ge X_{(2)} \ge \ldots \ge ...
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1answer
38 views
i.i.d assumption: formal definition vs. intuition [duplicate]
Intuition
In ML, as I constantly run into the i.i.d assumption for datasets, I have an intuition of what this assumption really means. So if I'm not mistaken:
"independent" means that ...
5
votes
2answers
55 views
If $X$ is a continuous random variable with support $A$, does this imply that the cdf of $X$ is strictly increasing on $A$?
If $X$ is a continuous random variable with support $A$, does this imply that the cdf of $X$ is strictly increasing on $A$?
My guess is yes. But just in case, let me know if you can think of any ...
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vote
2answers
50 views
Estimator of $Y$ in the simple linear regression model
In a statistics textbook I saw that the linear simple regression model is defined as
\begin{equation}
Y = \alpha + \beta x + e
\end{equation}
where $x$ is a value of the independent variable, $Y$ is ...
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0answers
34 views
Random number (between 0 & 1; > 5 decimal places) from skewed binomial/beta-like distribution, with set mean (same as mode) and set variance
Disclaimer: I already asked this question once, however it was misunderstood and I was told by a moderator to repost it: Random number (between 0 & 1; > 5 decimal places) from binomial/beta-...
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0answers
98 views
Conditions for gaussianity of sum of Gaussians [duplicate]
Under which conditions is the sum of a finite number of Gaussian random variables a Gaussian?
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29 views
What conditions are needed for $a_n = O_p(n^d) \implies E[a_n] = O(n^d)$?
Let $X_n$ be a uniformly integrable sequence of random variables. In a recent question I asked about the possibility of converting Big $O_p$ convergence in probability of the sequence $X_n$ to Big $O$ ...
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0answers
25 views
joint PDF of continuous and discrete random variables
Given exponential a random distribution X with PDF $f_X(x)=\lambda e^{-\lambda x}$ and a random variable Z with the PMF $p_Z[z]=0.5, z= \pm1$, I am trying to find the PDF of $Y=ZX$ (I also know that Z ...
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1answer
110 views
Can we go from $X_n = \mu + O_p(n^{-1})$ to $E[X_n] = \mu + O(n^{-1})$?
Let $X_n$ be a uniformly integrable (UI) sequence of random variables. If we have
$$
X_n = \mu + O_p(n^{-1}),
$$
then for $0 \le \delta < 1$ this implies
$$
X_n = \mu + o_p(n^{-\delta}) \quad \quad ...
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votes
2answers
55 views
Why does the superposition of two processes with geometrically distributed interarrival times not result in a process with similar distribution?
Assume we have two random variables $G_1$ and $G_2$ that both follow a geometric distribution of type B (so RVs can assume 0) with success probability $p = p_1 = p_2$ that describe the interarrival ...
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2answers
289 views
Random number (between 0 & 1; > 5 decimal places) from binomial/beta-like distribution, with set mean (same as mode & median) and set variance
Due to misunderstandings and as per request I have rollbacked this question to this previous state.
Please do not answer this question instead answer this one: Random number (between 0 & 1; > 5 ...
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0answers
14 views
Calculate Conditional Expectation using dataset and covariance matrices
So I'm having trouble understanding what I'm doing wrong here.
For context, I have some velocity components in my dataset for turbulence (simplified).
I have flattened them out so my 3 velocity ...
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0answers
13 views
Probability distribution of training set t in Bishop's pattern recognition book
I'm currently struggling with understanding the Bayesian approach to machine learning. Which is one of the paradigms presented in Bishop Pattern Recognition and Machine Learning. Since some parameters ...
4
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1answer
60 views
Is $\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ uniformly integrable (UI)? What assumptions make it UI?
$\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$
Let $\mathbf{X}_n$ be the usual data matrix in standard multiple regression where I have used the subscript $n$ to indicate the number ...
3
votes
2answers
89 views
When are these functions of a random variable independent?
Assume that $I$ is an indicator variable
\begin{equation}
I=\begin{cases} 1 &, \text{if} \,X<0 \\ 0 &, \text{else}\end{cases}
\end{equation}
and $X$ is a random variable. I want to know if $...
4
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1answer
74 views
Separating $X$ from $Y$ in $E[(X^T Y))^p]$ for $p = 3$ and $4$?
Let $X$ and $Y$ be random vectors with $X$ independent of $Y$. When dealing with terms of the form $E[(X^T Y)^p]$ for $p \ge 1$, it is very useful to be able to separate the $X$ from the $Y$.
For $p=1,...
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3answers
68 views
Is it possible to interchange the quantile operator and a measurable monotone function? $Q_\theta(f(X)) = f(Q_\theta(X))$
Let $Q_\theta(X)$ is the $\theta^{th}$ quantile of a random variable $X$, and if $f$ is a measurable strictly increasing function.
I want to know if $Q_\theta(f(X)) = f(Q_\theta(X))$.
I know that for ...
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0answers
25 views
Coskewness: three or two random variables?
The wikipedia for coskewness says
coskewness is a measure of how much three random variables change
together
It then says
If two random variables exhibit positive coskewness they will tend to
...
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0answers
9 views
Covariance between parameters from different regression models
Is there a formula for the covariance between regression slopes from different models, fitted to the same data? For example, if I have a finite and fixed sample, $S$, and models:
$Y = b_1X$
$Y = b_2X ...
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0answers
16 views
Bounding the moments of the argmax of continuous process
I need to calculate/upper bound the second moment of the variable
$t^{*} \triangleq \underset{t>\alpha}{argmax} \{W(t) - t^2\}$ where $W(t) \triangleq B(t) - B(t - \alpha), \alpha \in R^{+}$ and $...
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0answers
24 views
What's the term for a r.v. X “upper bounding” Y probabilistically? [duplicate]
What's the term for when a random variable $X$ has a higher probability of being greater than t than the probability of $Y$ being greater than $t$ for all real $t$? I recall reading a term for this ...
3
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1answer
88 views
Finding the probability that an exponential random variable is less than a uniform random variable
I have the following statement I am trying to solve: Let $T$~ $exp(2)$ and let $X$ ~Uniform$(0,1000)$ , where $T$ and $X$ are independent random variables, what is $P(500 + 1200T - X < 0)$ and ...
3
votes
1answer
48 views
Variable transformation under a discontinuous mapping function introduce different bias?
For any real number $x \in R$, consider its randomized version $y \sim \mathcal{N}(x,1)$. Now consider a mapping function of $y$ denote by $f(y)$. We want to study the following bias term $R(x) = | E[...
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1answer
52 views
How does a pdf change after a variable transformation with another random variable?
I have a probability density function of the energy $f(E)$ of a distribution of particles. Now, each energy gets shifted according to an angle $\theta$: $$E_{after} = E_{before} + g(E_{before}) \cos \...
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0answers
10 views
Strict Sense Cyclostationary and Shifting the X with $\theta$
Fellow stackexchangers, I did my best to put a topic that describes the question that I am going to ask. I am reading Probability, Random Variables, and Stochastic Processes by Papoulis and I am ...
2
votes
1answer
258 views
Show that maximum of two random variables is a random variable
If we have that $X$ and $Y$ are random variables, how do we prove that $Z=max(X,Y)$ is also a random variable ? I want to do this by showing that $Z$ is measurable, but I don't know how to do this.
2
votes
1answer
82 views
Random variable without pdf but with a cdf?
In this video, the professor says that some random variables have no pdf but do have a cdf. Also, in my course material, I studied that converging in mean was stronger than converging in cdf which ...
2
votes
1answer
33 views
When is Jensen's Inequality strict?
For a homework problem, I have to prove that for a random sample $X_1, \ldots, X_n$, drawn from a population with finite variance $\sigma^2$, with sample mean $\bar{x}$ and sample variance $s^2$, that ...
