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).

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Taking a limit under absolute value

Assume $Z = \mu V + \sigma \sqrt{V}U$, where $V \sim \Gamma(n/2,1/2)$ and $U$ is a sum of dependent standard normal random variables. I'm looking at $\frac{|Z|}{n}$, as $n \to \infty$. My question: is ...
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Why can a set of random variables be called a sample? [duplicate]

I am trying to understand this slide on sample means. I understand the meaning of the word sample to be a set of individual objects as if grabbing a handful of jellybeans from a packet. Yet in this ...
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I can't find a way to find the mean of x and the standard deviation of x of this problem [closed]

I can't find a way to find the mean of x and the standard deviation of x in this problem without making an insane amount of calculations. The problem is below. Please help. When planning a party Mr. ...
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Sum and Multiplication of Two Normal Distributions [duplicate]

I'm not sure about how to go about this question. How do we deal with the variances when the two normal variables are summed?
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Can a random variable be uncorrelated with its product with a correlated random variable?

I have a random variable $X.$ I want to find a random variable $Y$ such that $Y$ is correlated with $X,$ but $Y$ is not correlated with the product of $X$ and $Y.$ Is it always possible?
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How to estimate correlations between products of random variables?

I have three random variables: A, B and C. I know their pairwise correlations. In other words, I know what is correlation between A and B, B and C, and finally, A and C. I also know means of all three ...
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Likelihood $L(\theta; \mathbf{y})$: Is $\theta$ a vector of parameters or is it a single parameter?

I have the following definition of likelihood: Let $y_1, \dots, y_n$ be a sample of observations taken on corresponding random variables $Y_1, \dots, Y_n$ whose distribution depends on the parameter(...
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How are differential equations and stochastic differential equations different?

In the simplest terms, how are differential equations and stochastic differential equations different? As far as I can tell, SDEs are PDEs or ODEs, where the derivative of some function wrt itself is ...
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Projecting a distribution onto an arbitrary norm

So we represent the Dirichlet distribution as the projection of the $d$ independent gammas (on $R_+^d$ onto the unit simplex, and we arrive at that through the $L_1$ norm. That is, divide ${\bf x} \...
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Mapping a wrapped Cauchy distribution to a uniform distribution?

I'm investigating model mismatch and have a wrapped Cauchy distribution of f(x,p) = (1-p^2)/ (2*pi*(1+p^2-2p*cos(x))) Is there a way to map this to a random ...
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How to standardize uniform variates?

How do you standardize a set of uniform variates on the interval (0,1) to have mean 1/2 and variance 1/12, while staying in the interval (0,1)? The usual procedure of shifting and scaling variates to ...
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Change of metric for probability density vs for probability

When one changes the variable in a probability density function, one must account for the jacobian to ensure the elementary probability is constant (eg Derivation of change of variables of a ...
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Understanding Rayleigh Distribution

I am an aerospace enthusiast. I have obtained wind speed data that's been gathered and published by NCEP/NCAR. It provides wind speed data for every $2.5^\circ$ increment in latitude and longitude for ...
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Conditional PDF given a random variable theory

I am trying to understand the derivation of the formula for a conditional pdf given another random variable, but I am unsure if my interpretation is correct. I am following the course from MIT - ...
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1answer
26 views

Convergence to 0 in probability for non-iid random variables

Assume $U_k$ are correlated standard normal random variables. Let $R_k := a_k U_k^2$, with $a_k > 0$ and $\sum_{k=1}^{\infty} a_k < \infty$. How can we prove that $S_p:= \frac{1}{p}\sum_{k=1}^{p}...
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Can someone help me understand the random effect parameters in my linear mixed model output?

I have some data for which I modeled in a linear mixed model. I understand everything except the random effect parameters. These variance parameters appear to be bound between -1 and +1. How do I ...
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If $E[|X_n|] = O(n)$ is $E[|X_n|^2] = O(n^2)$?

Let $X_n$ be a random variable that depends on $n$ and suppose $E[|X_n|] = O(n)$. Then can we say $E[|X_n|^2] = O(n^2)$? If it doesn't hold in general, are there particular interesting cases where it ...
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Question about random sample vs support

I got a quick question. Say you have X is random variable X=1 you have sucess X=0 you have failure. And you the following list of number [1,0,0,0,1] So would the list be the support of the random ...
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1answer
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CLT for non iid random variables

Assume $U_k$ are correlated standard normal random variables. Let $R_k := a_k U_k$. I'm looking for CLT of the sum $S_p := \sum_{k=1}^{p}\frac{R_k}{\sqrt{p}}$. Since $U_k$ are correlated, I'm looking ...
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Sum of correlated squared normals

Assume that $(X_1,X_2)' \sim \mathcal{N}((\mu_1,\mu_2)', \Sigma)$, $j =1,2$, and $Cov(X_1,X_2) = r > 0$. We know that $X_1 + X_2 \sim \mathcal{N}(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2 + 2r)$. ...
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Calculating the sample cumulative distribution function for a Kolmogrov-Simulation test to examine the goodness of fit with given data [duplicate]

I have sample data for 'Times between successive crashes of a computer system' which is for a 6 month period and the data is given in hours. The data in brief is : 1,10,20,30,40,52..... I need to use ...
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Covariance between the linear combination of lognormal random variables

I have two lognormally distributed random variables $Y_i=e^{X_i}$ where $X_i \sim \mathcal{N}\big(\mu_i, \: \sigma_i^2 \big)$ for $i=1,2$, and $X_1$ and $X_2$ are correlated by $\rho_{12}$. Now, Let $...
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Does mutual independence of X, Y, Z implies conditional independence of X and Y, given Z

Given mutual independence of 3 r.v.s X, Y, Z, can we conclude that X and Y are independent, given Z? Note that I am interested in case when all 3 r.v.s are mutually independent, not only pair X, Y. In ...
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Is an Expected Value Fixed or Random

Suppose I have a random variable, x, and then let E(x) = p denote its expected value. I believe p can be treated as non-stochastic (fixed), but the variable x obviously cannot. Is that assumption ...
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Proof for X and Y independent if joint c.d.f. of both variables is the product of c.d.f of each of the two variables

The book I am reading says the following: For any two variables $X$ and $Y$, if for every two sets $A$ and $B$ of real numbers $Pr(X \in A \cap Y \in B) =Pr(X \in A )Pr( Y \in B)$, then $X$ and $Y$ ...
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Clarification regarding $(X'BX)(X'AX)$ distribution

Assume $X = (X_1, \ldots, X_n)' \sim \mathcal{N}(0, \Sigma)$ is a random normal vector. I'm looking for the distribution of the following form: $$Z = (X'BX)(X'AX)$$ here $X \in \mathbb{R}^{n \times 1}$...
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Distribution of quadratic expressions

Reading similar questions for the quadratic form of normal random variables(with the derivation here), I'm interested in how well can we generalize this approach for other distributions? E.g., if $W = ...
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Distribution of $X_1^2 + X_2^2$ and $X_1^2 X_2^2$ for correlated normal r.v

Assume that $X_1, X_2$ are standard normal random variables with $Cov(X_1,X_2)=a$. Then $X_1^2, X_2^2$ are correlated gamma random variables. Are there any known results for the distribution or the ...
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Joint cumulative distribution

I'm studying joint distributions of two random variables, $X$ and $Y$. Ross's book (Chapter 6) defines the joint CDF as $F(a, b) = \mathbb P(X \le a, Y \le b)$ and the PDF as $f(a,b) = \frac{\partial}{...
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Finding $Cov(X_1^2, X_2^2)$

Assume $X_1, X_2$ are dependent random standard normal variables with $Cov(X_1,X_2)=a$. What is then the $Cov(X_1^2, X_2^2)$? Are there known results for this without going technical into $\mathbb{E}...
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1answer
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Need help with understanding random variables/the data generating distribution

Lets say we want to predict a persons weight using their height and gender. We always assume there is a data generating distribution $P_{X×Y}$, and all output and input pairs are generated i.i.d from $...
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Does the sum of two variables second order stochastic dominate the mixture of two variables

hello I have two independent variable P and Q. They are both non-negative. Let $\alpha \in (0,1)$. Now I define two new variables on them: The first variable is the sum of the two variables $$R_1:=\...
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1answer
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Stopping Time for a sequence of Random Variables

If $M$ and $N$ are stopping times of sequence $\{X_n\}$ where $n \geq 1$. Then are $min (M, N)$ and $max(M,N)$ also stopping times of the sequence $\{X_n\}$ ? Is there any rigorous way to prove ...
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In Choce Experiments (DCE) creating cards from scenario sets must be random?

It will be difficult for me to describe it in one sentence, I will use an example. I would like to know if I can arrange the cards in the block combining scenarios other than randomly. In Aizaki, H., &...
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1answer
129 views

What is the expected value of the division of a random variable by a sum of random variables?

With $X_1$, $X_2$ and $X_3$ being independent random variables, how can I compute $\mathbb{E}\left[ \frac{X_1}{X_1+X_2+X_3}\right]$? Is $\mathbb{E}\left[ \frac{X_1}{X_1+X_2+X_3}\right] = \frac{\mathbb{...
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1answer
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Variance involving two independent variables

hello I have two independent variable P and Q. They are both non-negative. Now I define two new variables on them: The first variable $$R_1=\alpha P+(1-\alpha)Q.$$ Since P and Q are independent, so $$...
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Suppose X1∼U(0,1) and X 2|X1 =x1 ∼U(0,x1) are uniform random variables. Compute probability of (X1+X2≥1)

The answer to this problem is (1-ln2). I am getting 0.5 which is not even close. Any kind of hints or even suggestive reading would be helpful as I am getting a lot of doubt in problems of the same ...
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Expect hitting time of a discrete time random walk with complex step size distribution

Suppose a random walk starts from $S_0=0$. The iterative equation is $$S_{t+1}=\max\{S_t+y_{t+1}-k,0\},$$ where $k$ is a fixed value that is larger than 1, and $y_t$, $t=1,2,\cdots$, are i.i.d. and $$...
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Finding the Probability of a Conditional Normal Distribution

I'm trying to answer the following question: Suppose that $X$ and $Y$ are independent normal random variables with mean $\mu_1$ and $\mu_2$ respectively, and variance $\sigma^2_1$ and $\sigma^2_2$ ...
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1answer
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Covariance between a binomial random variable and its size (number of trials) (found in the context of binomial thinning)

Assume we have a random variable $X$, and we construct another random variable $Y$ to be from a binomial distribution of size $X$ and success probability $\alpha$, i.e., $Y \sim Binom(X, \alpha)$. How ...
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questions about random variables, if the following statement is true and why

questions about random variables, if the following statement is true and why
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Why use complex-valued random variables?

Edit: This question has been posted on Math.exchange here. To avoid duplication, please comment on the Math.exchange thread. I am interested in random complex numbers and am trying to understand why ...
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1answer
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Expected Value of the Ratio of Independent Variables, E(X/(X+Y)) [duplicate]

If $X$ and $Y$ are independent random variables, is the following true? Is there an easy way to show this? $$E\left[\frac{X}{X+Y}\right]=\frac{E[X]}{E[{X+Y}]}=\frac{E[X]}{E[X]+E[Y]}$$ If this is not ...
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1answer
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Way to define marginal probabilities

I found this notation in one paper that focuses on copulas: Consider a $d$ -dimensional continuous random vector $X=\left\{X_{1}, X_{2}, \cdots X_{d}\right\}$ with marginals $F_{i}\left(x_{i}\right)=$...
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Can we derive the Variance of the least squares slope WITHOUT assuming that $X_i$s are fixed or deterministic? [duplicate]

Everywhere in the literature, I have seen that while deriving the variance of the least squares slope estimate $Var(\hat \beta_1) = \dfrac{\sigma ^2}{SS_{xx}}$, we always assume that $X_i$s are fixed ...
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Mean preserving spread and truncated distributions

Take two distributions $F_B(x)$, $F_A(x)$ with the same support. Assume that B is a mean-preserving spread of A. What I want to understand is whether $E_{A}[x | x \leq t] \geq E_{B}[x | x \leq t]$, ...
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239 views

Independence of variables in expectation

I know that $X$ and $Y$ are independent, and have an expression $$E[I(Y>X)*I(X>2)].$$ Is the independence between $X$ and $Y$ enough to say that $$E[I(Y>X)*I(X>2)] = E[I(Y>X)]*E[I(X>...
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How to understand the difference between the mixture of same distributions vs. convolution (sum) of random variables of same distributions?

Before I ask the question, let me introduce how I came to this problem. Recently I learned about the linear regression. It was said, that the residuals of the model should be normally distributed. We ...
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1answer
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Apparent contradiction between expectation of the product of random variables and law of total expectation

Suppose we have two random variables $X, Y$. Then, in general, if they are dependent $$E[XY] \ne E[X]E[Y]$$ However, according to the law of total expectation, $$E[XY] = E_Y[E_X[XY|Y]] = E_Y[YE[X]]=E[...
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1answer
64 views

Generalized variance of the sum of N correlated random variables

I am trying to model the variance of a time series $Y_n$ which is the sum of $n$ observations of $X_i$. I've reviewed the other answers on CrossValidated; however, I haven't been able to apply those ...

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