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
0answers
13 views

Random right cenoring

I observe a series of values from different trials ($Y_1, ... Y_N$). All values come from the same distribution ($F(\cdot)$). Trials do not have the same number of values ($N$ differs across ...
0
votes
0answers
24 views

Generating Poisson random Variable [on hold]

Please help me in correcting the following code for generating 3 Possion random variables. n=3 lambda=.7 p=exp(-lambda) f=p set.seed(34231) x <-0 for(i in 1:n) { U=runif(1) show (p) show (f) ...
1
vote
1answer
15 views

Minimizing MMSE over positive random variables

Let X be a random variable with a finite second moment. We know that: Argmin E(X-Y)^2 = E(X|g), Where the minimum is taken over all g-measurable random variables Y. How can I find argmin E(X-Y)^2 ...
0
votes
0answers
5 views

Why is the parameter and the random variable swapped in this conjugate distribution pdf?

I'm reading a journal titled Claims reserving in the hierarchical generalized linear model (hglm) by Gigante, Picech and Sigalotti. In the distributional assumption for the unobserved risk parameters, ...
8
votes
1answer
342 views

Is the sum of a large number of independent Cauchy random variables Normal?

By Central Limit Theorem, the probability density function of the the sum of a large independent random variables tends to a Normal. Therefore can we say that the sum of a large number of independent ...
1
vote
3answers
43 views

Prove that this doesn't converge almost sure to 0

Suppose we have $X_n$ a random variable, that can take two values: $X_n = \begin{cases} 0, & \text{with probability 1 - $\frac{1}{2n}$,} \\ n, & \text{with probability $\frac{1}{2n}$} ...
0
votes
0answers
16 views

Treating X as non-random under random sampling

Random sampling allows us to treat values of independent variables that have been sampled as non-random (for the purpose of proving the unbiasedness of OLS estimators). How is this possible? My ...
0
votes
0answers
20 views

Conditional expectation of random vector given one variable

I would like to find the conditional expectation $E(Y_i|X_i)$. Here $Y_i=\mu+\beta X_i+\gamma X^{T}+\epsilon_i$ where $X=(X_1,X_2,\ldots, X_n)$ is a $n\times 1$ vector, $X_i$ is iid random variable ...
0
votes
1answer
23 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 ...
1
vote
1answer
43 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 ...
0
votes
0answers
26 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 ...
1
vote
0answers
6 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 ...
0
votes
3answers
36 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 ...
0
votes
0answers
22 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 ...
0
votes
0answers
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: ...
1
vote
3answers
131 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 ...
1
vote
0answers
53 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 ...
4
votes
1answer
108 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, $X_1$, $X_2$, ...
5
votes
3answers
97 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 ...
1
vote
0answers
34 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 ...
0
votes
0answers
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 ...
0
votes
0answers
30 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 ...
0
votes
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 ...
5
votes
1answer
192 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 ...
1
vote
1answer
29 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 ...
1
vote
0answers
16 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 ...
3
votes
0answers
46 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 ...
1
vote
2answers
73 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 ...
1
vote
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 ...
1
vote
0answers
25 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 ...
0
votes
0answers
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
1
vote
1answer
25 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 ...
2
votes
2answers
205 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 ...
0
votes
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 ...
1
vote
0answers
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 $x$, ...
1
vote
1answer
78 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 ...
0
votes
0answers
26 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 ...
1
vote
1answer
118 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 ...
2
votes
0answers
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 ...
1
vote
1answer
52 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 ...
2
votes
0answers
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 ...
0
votes
1answer
66 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 ...
1
vote
0answers
43 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 ...
1
vote
2answers
51 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
votes
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 $\hat{\sigma}$ ...
0
votes
0answers
38 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
votes
1answer
31 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 ...
1
vote
2answers
131 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
votes
1answer
46 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
votes
1answer
91 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 ...