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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20 views

Transformation from skewed to symmetric distribution

Let us consider a positive valued random variable $X$ which is following a positively skewed probability distribution. Is it possible to a get a function $f$ (one-to-one) for which $f(X)$ follow a ...
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1answer
27 views

Exemple of real-life non-linear correlation in time series?

I am looking for a dataset of 'real-life' time series that exhibits non-comonotonic / non-countermonotonic dependence. I am not looking for the textbook X^2 correlation, but interesting yet real ...
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22 views

Difference between (quantitative,qualitative) vs (discrete,continuous)

Could someone please clarify the difference between these seemingly interchangeable groups? (quantitative,qualitative) vs (discrete,continuous). Obviously they are not, but I cannot seem to pinpoint ...
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4 views

Calculating joint entropy of two variables having null values

I calculated the joint entropy of two random discrete variables by zipping their values in a 2-tuple and applying the formula: where n is the number of distinct classes and q is the probability of ...
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3answers
30 views

Square of the Sample Mean as estimator of the variance

Suppose we have the following random variables $X_1$, $X_2$,....$X_n$,.., that are $iid$ but we dont know what distribution they follow. I know that the sample mean $\bar{X}$ is an unbiased ...
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19 views

Best random variable for infinite trials of a true/false event?

if you were to toss a fair coin a finite amount of times, and a success = heads, then the best random variable to represent it would be a binomial random variable. However, if you were to toss it an ...
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12 views

Computing SD for function of two RV's in R gives wrong results [duplicate]

I have 2 random variables, all independant, discrete and uniform I want to compute standard deviation for Z = X - Y Here is the R code: ...
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3answers
126 views

Decrease of $(X'X)^{-1}$ as n increases

Let $X$ be a $n \times p$ matrix, filled with iid draws, with $n \geq p$ (like a conventional data matrix). I would like to show that, in a sloppy notation, $(X'X)^{-1} \rightarrow 0$ as $n ...
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50 views

Distance between random variables [closed]

I have found plenty of ways to compute the distance between random variables. However, I did not find any taking something else than the random variables as input. Do you know whether or not there is ...
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1answer
103 views

What is the domain of this random variable?

I've been self-studying Introduction to Statistical Learning. From page 16 of the book: "...suppose that we observe a quantitative response $Y$ and $p$ different predictors, $X_1$, $X_2$, ...
2
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1answer
63 views

Expectation of rational formula

I have two independent normal random variables $x$ & $y$ that are zero mean and unit variance. $a$ & $b$ are positive. I need to find the mean of $$z=\frac{ax^2y^2}{1 + bx^2}.$$ Any help ...
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33 views

Splitting up the variance of Z for Z = X*Y

$Z$ is a function of two dependent random variables, e.g. $X \cdot Y$. Here it is shown that $$var(Z) = var(XY)=(cov(X^2,Y^2)+E[X^2]E[Y^2])-(cov(X,Y)+E[X]E[Y])^2$$ I am interested in a metric that ...
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31 views

Modeling the joint distribution of stream statistics

I have a question regarding computing the joint discrete probability distribution of statistics in a number stream. I posted this problem in the Mathematics section as well but I'm hoping the ...
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0answers
29 views

Probability of X > Y and X > 0 (Frechet case)

I have two random variables, $X$ and $Y$. (They are both Frechet with different $\sigma$ parameter, but I'm interested in the general probabilistic argument here as well, so generality of answers is ...
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0answers
26 views

Converge in Probability of random variables

I don't know if I understood the (Convergence in probability of random variables) formula, why $ | Xn -X | $ should be $$ \geq \varepsilon $$ For example if $\varepsilon $=5( a random number), and ...
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191 views

Difference of Frechet variables

Let $$ X \sim Frechet(\alpha, s_1, m)\\ Y \sim Frechet(\alpha, s_2, m) $$ I'm trying to compute $Prob(X > Y$). This is equivalent of computing $Prob(X - Y > 0)$. Unfortunately, this is where ...
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1answer
25 views

Expectation of $\mathbb{E}(Tr(X^T A X))$ and $Var(Tr(X^T A X))$?

What is the expectation: $\mathbb{E}(Tr(X^T A X))$ and $Var(Tr(X^T A X))$ when $X_{i,j} \sim N(\mu, \sigma^2)$ and $X \in \mathbb{R}^{n \times k}$ where $n>k$ and $A$ is a given p.s.d matrix (not ...
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15 views

Sum of uniformly distributed random variables over different intervals?

Let $\{X_i\}_{i=1}^N$ be $N$ random variables uniformly distributed over the intervals $[a_i, b_i]$ respectively. How does the sum: $$\sum_{i=1}^N X_i$$ distribute? This is a generalization of the ...
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42 views

How to check if functions of i.i.d random variables are dependent or independent?

i'm new to this forum and the science of statistic.This is my question: Let's say that we have two i.i.d random variables X and Y, which both follow a Rayleigh distribution. Then, we define two new ...
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2answers
63 views

Lower-bound for total-variation between two sub-gaussian random variables

Problem setup Let $X$ and $Y$ random variables on the real line with following properties: $X$ and $Y$ are $\sigma^2$ sub-gaussian $E[X] = 0$ and $E[Y] = \Delta$. Question Whats the lower ...
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1answer
33 views

Check a sortition

[Warning: newbie in stats. It's a practical problem at work, not a homework, I'm not a student.] I have N entities and a process chooses M randomly (in theory...). M < N If there are two ...
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24 views

Finding conditional expected value

Given that X and Y are two independent exponentially distributed random variables with parameters a and b respectively. let Z = max(X,Y) find E[X|Z] attempt: I found that: P(Z=X) = b/(a+b) and ...
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33 views

Can you explain why this distribution has been suggested for the length of a telephone call

I don't understand what the distribution in e) shows me? I would expect a normal curve about the mean but this is confusing
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1answer
24 views

Double Partial Derivatives of sum of variances of dependent random variables

I have the following function $$f(α)=Var[αX+(1−α)Y]=Var(αX)+Var[(1−α)Y]+2α(1−α)Cov(X,Y)$$ Partial derivative of this function w.r.t α leads us to following result ...
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196 views

How does Slutsky's theorem extends when two random variables converge to two constants?

The 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 element $X$ and $Y_n$ converges in ...
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29 views

Intuitive meaning of IID sample [duplicate]

I am a software engineer, who recently started self learning statistics as per the requirements of a personal project which involves time series analysis. With the goal of quickly gaining a basic ...
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1answer
29 views

Expectation of a function of a binomial distribution

I have a question that is: Given n iid Bernoulli(p) distributions: $X_1, X_2, \ldots, X_n$ and $S_n=\sum X_i$. Find $E[(S_n-np)^3]$. Hint: $S_n-np= \sum (X_i-p)$. So far, I have gotten that $S_n$ is ...
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96 views

Functions of random variables: relationship between means

Let $x$ be a continuous random variable and $y=f(x)$. The distribution of $x$ is not known. If so, (a) is there a way to find out the mean of $y$, say $\bar{y}$, as a function of the mean of $x$, ...
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1answer
76 views

Finding Probability Mass Function (PMF) Given a Geometrically Distributed Random Variable and a Negative Binomial Random Variable?

I am a non-student working through the first edition of Yates and Goodman's text, Probability and Stochastic Processes. On page 115, question 3.6.9 goes like this: Each millisecond at a telephone ...
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25 views

Modeling a time series - help in understanding the approach in a paper

The question is based on a paper titled : Forecasting high waters at Venice Lagoon using chaotic time series analysis and nonlinear neural networks On page 2 right above Eq(1), the authors say ...
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1answer
77 views

How do I compare the the sampling distribution of the minimum of a distribution by sample sizes

I saw this question (link) but when I read it, I see that it has a fixed "N" so I thought it was asking about for a finite sample size. When I read the answer that it was suggested to be a duplicate ...
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23 views

Sampling from a distribution using a variable transformation

I would like to draw samples from a pdf $f$. I can write $f$ as $f(x) = A g(y(x))$, where $A$ is a constant and $g$ is another pdf from which I can easily draw samples. Is there a way to first draw ...
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1answer
51 views

Intuition for Identically Distributed Random Variables

How do I successfully communicate an intuitive understanding of what it means for two random variables to be "identically distributed"? The definition is easy to state, but it lacks insight and why it ...
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53 views

How is the minimum of a random set of random variables distributed? [duplicate]

If $X_1,...,X_N$ are independent and identically distributed exponential random variables, what can be said about the distribution of $\text{min}(X_1,...,X_N)$ when $N$ is random and modelled as a ...
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61 views

Categorical or Quantitative?

A colleague and I had a conversation about whether the following variables are categorical or quantitative. 1) Social security numbers 2) Phone numbers 3) Postal zip codes We agreed that all three ...
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41 views

Proof of almost deterministic random variables [closed]

Let $X$ and $Y$ be independent random variables and suppose that $P(X + Y = c) = 1$, where $c \in \mathbb{R}$ is a constant. Prove that $X$ and $Y$ are both almost deterministic random variables ...
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2answers
48 views

Probability of $X^2$ (random variable) using pchisq() function in r [closed]

How to find the probability of a random variable $X^2$ using pchisq() function in r? $p\,(1<X^2<2)$ How to find the probability of this random variable ...
2
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1answer
35 views

Working out the expectation of a function of iid random variables

I have found the maximum likelihood estimator $\hat{\sigma}$ of a iid r.vs $X_1, ..., X_n$ which all have normal distribution with known mean $\mu$ and unknown variance $\sigma^2$. So $\hat{\sigma}$ ...
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36 views

How to sort a set of data to get a particular correlation

I have a set of data $(X,Y)$, where $X=1,2,3....100$ and $Y$ is a set of values drawn from a lognormal distribution. I would like to sort the values of Y such that the sorted data $Y_{sorted}$ has a ...
3
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1answer
29 views

Widgets and boxes problem: expectation and variance. Why is this wrong?

I'm taking the MITx: 6.041x Introduction to Probability - The Science of Uncertainty class to sharpen my probability skills. In one of the problems, the solution I came up with diverged from the ...
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2answers
123 views

Why is $P(\min\{X_1,…,X_n\} ≥ y)=P(X_1≥y,…, X_n≥y)$?

Why is $$P(\min\{X_1,...,X_n\} ≥ y)=P(X_1≥y,..., X_n≥y)$$ and similarly $$P(\max\{X_1,...,X_n\}≤y)=P(X_1≤y, ..., X_n≤y)$$ I.e. why are $\min$ and $\max$ equivalent to AND probabilities of all the ...
0
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1answer
44 views

An entropy and mutual information problem

Let's suppose we have 4 random variables X,Y,Z and T and that the following equations hold about the entropy: $$H(T|X)=H(T)$$ $$H(T|X,Y)=0$$ $$H(T|Y)=H(T)$$ $$H(Y|Z)=0$$ $$H(T|Z)=0$$ I want to prove ...
3
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1answer
89 views

“Let random variables $X_1,\dots, X_n$ be a iid random sample from $f(x)$” - what does it mean?

In books it is often written, Let random variables $X_1,\dots, X_n$ be a iid random sample from $f(x)$. What does it mean? Are $X_1,X_2,\dots,X_n$ different values of one random variable $X$ which ...
2
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0answers
19 views

zipf and correlated lognormal

I have been struggling with this for a while. I want to generate two random variables $X$ and $Y$ with a particular correlation $\rho$ where $X$ is the file popularity (zipf distribution) and $Y$ is ...
3
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2answers
126 views

Probability density of sum of two beta random variables

Suppose that $X$ and $Y$ are independent and have beta distributions. $X$ has probability density function $g(x)=6x(1-x)$ for $ 0\leq x \leq 1$ and $Y$ has the probability density function ...
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56 views

Determine distribution of a limit of random variables

Suppose a box contains a blue ball and green ball. After every hour a ball is chosen from the box randomly and then is put back with another ball of the same colour. After $h$ hours, there are ...
3
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1answer
134 views

Using the correct discrete random distribution for a Conditional PMF problem: “Suppose you arrive at a bus stop at time 0…”

I am a non-student working through the first edition of Yates and Goodman's text, Probability and Stochastic Processes. On page 115, question 3.6.8 goes like this: Suppose you arrive at a bus ...
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29 views

the functions of random variables

I'm studying for a course and a question came up asking to find multiple functions h such that $Y=h(X)$, where $X \sim$ uniform$[0,1]$ and $Y \sim$ uniform$[-2,8]$. I understand that $f(x) = 1$ when ...
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37 views

Probability Density Function of a linear combination of 2 dependent random variables, when joint density is known

Let's say there are two dependent random variables $X$ and $Y$ with joint density function $f$. What is the PDF of the weighted sum of these two variables, $Z = aX + bY$? Thanks in advance for any ...
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34 views

Probability Density Function of the sum of N random variables [duplicate]

I have N random variables: $X_{1}, X_{2}, ... X_{n}$ which are all independent. The probability density function for the ith random variable is $f_{i}(x)$. What is the pdf $f_{X}(x)$ where ...