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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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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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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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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How to calculate the sample cumulative distribution function for a Kolmogrov-Simulation test to examine the goodness of fit with given data?

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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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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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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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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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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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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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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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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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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Discrete Random Variable Confusion

This seems like a simple question, but I am unsure. Please bear with me and thanks for the help. I am told to suppose that A,B are discrete random variables that have a joint pdf, and am told to ...
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Random variable with finite logarithmic first moment, infinite logarithmic variance [duplicate]

Could you provide an example of a random variable $X$ such that $|\mathbb{E}(\ln(X))|<\infty$ but $\text{Var}(\ln(X))=\infty$, if such a random variable exists at all? Related: "Random ...
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Random variable with finite exponential first moment, infinite exponential variance [duplicate]

Could you provide an example of a random variable $X$ such that $\mathbb{E}(e^X)<\infty$ but $\text{Var}(e^X)=\infty$? Related: "Random variable with finite logarithmic first moment, infinite ...
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Binomial-like distribution which the probability changes after certain trials

Let $0<p<q<1$. Consider a trial with success probability defined below. If it is a $100, 200, 300$th trial after the last success, the probability is $q$. If it is a $400$th trial after the ...
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conditional probability with multiple random variables

The rule of conditional probability states: $$ P(A|B) = \frac{P(A,B)}{P(B)} $$ However, it is not clear to me how you can/must condition the random variables with more than just two. The top answer on ...
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Determining independence (dependence) normal random variables

I've the following question which I'm struggling to deduce independence (for part b and c) based on theorems given in the class for the multivariate normal distribution. I've the theorems, and managed ...
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2answers
157 views

Generation of random variables via composition and inversion

What are the main pros and cons of each method and when to use each one? Law [2007] mentions that: "Again, the reader is encouraged to develop the inverse-transform method for generating a ...
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Is it possible that marginally independent random variables are conditionally dependent?

Suppose that $X,Y$ and $Z$ are random variables. If $X$ is independent of $Z$ and $Y$ is independent of $Z$, is it possible that $X$ is dependent on $Z$ given $Y$ and $Y$ is dependent on $Z$ given $X$?...
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280 views

Is a “random variable” still a random variable if it is predictable?

If X in P(X) is considered a random variable because it varies among the occurrence the particular scenario which it may occur (such rolling a number on a die), can we still call X a “random variable” ...
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Finding Probabilities of Normally Distributed Random Variables [closed]

A random variable $x$ is known to follow a normal distribution with mean $35$ and standard deviation $7$ Find the following probabilities: a. $P(x<25)$ b. $P(x<33)$ c. $P(x>42)$ d. $P(x>35)...

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