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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Definition of random variable

In some books, they don't define the random variable based on measure theory. Instead, they define as follow (in the book All of Statistics of Larry Wasserman): My question is does this definition ...
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Mixed distribution of product of Bernoulli and Gaussian r.v

What should be the density function of the following mixed distributed random variable $Z$. $$Z \equiv X \cdot Y,$$ where $( \cdot)$ is product operation, and $X$ and $Y$ being independent, $X$ is ...
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R, Pairwise comparison of random variable in mixed model

We measured temperatures of a pond repeatedly every day at each hour for a month at two different depths (i.e., top and bottom). We want to see if the temperatures at the top of the pond are ...
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Square of gamma random variable [duplicate]

If i have a random variable with distribution $X \sim \Gamma(\alpha,\beta)$ then what would be the distribution of $Y = \lambda X^2$ (with $\lambda$ a scaling factor)? Can I say that $Y$ will follow a ...
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1answer
66 views

Inferring parent distribution using phase of 2D discrete fourier transform of random image

Suppose I have an NxM matrix with real entries $S_{n,m}$ which are iid with zero mean. Say $n \in [0, 1, ..., N-1], m \in [0, 1, ..., M-1]$. From $S_{n,m}$ I can compute the 2D discrete fourier ...
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49 views

Variance stabilisation

$Y$ has mean $\mu$ and variance function $V(\mu)$. If $V(\mu) = \alpha.\mu^v$ then $h(y) = y^{(2-v)/2}$ is variance stabilising which means that $Var(h(Y))$ is approximately constant. I tried to ...
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23 views

Advice on ANOVA formula for nested random design

I have read "Design and Analysis of Experiments" 8th Edition by D.C. Montagomery. In Chapter 14, there is a nested experiment with two factor A and B. Both factors are random. Then the ANOVA test ...
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1answer
36 views

Estimate variance of a function given variances of variables

I'm given the mean and the standard deviation of N random variables $A$, $B$, $C$, $D$... I compute the function $f:=f(A, B, C, D...) = \frac { AB... }{ CD... } $. How can I estimate the variance of ...
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182 views

Is it feasible to transform each variable differently while doing multiple regression

I have a dataset with 10 variables ...is it feasible to transform each variable differently while doing multiple regression... for example new_V1 = log(v1) New_V2= V2^2 New_V3= 1/V3 Likewise ...
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49 views

Bivariate distribution: beta and binomial

Consider a pair of RVs $X$ and $Y$, with the following conditional distributions: $$X | Y=y \sim Binom(L, y)$$ $$Y | X=x \sim Beta(\alpha + x, \nu)$$ where $L$, $\alpha$, and $\nu$; are all ...
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186 views

Using Uniform Distribution to Generate Correlated Random Samples in R

[On recent questions I was looking into generating random vectors in R, and I wanted to share that "research" as an independent Q&A on a specific point.] Generating random data with correlation ...
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Variance of the linear transformation of a random variable

I have a problem where the variance I'm calculating does not seem right. I have the following data: ...
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65 views

Generating and Working with Random Vectors in R

I've been trying to understand random vectors and generate them in R to reproduce properties. Recently, I asked a similar question, and it was rightfully placed on-hold for being too general. Thanks ...
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Generation of random variables [migrated]

I have a problem about the generation of random variables with R . I have to generate random variables $X_{ij}$ (i=1,...,25, j=1,...,5 ) knowing that each X_ij follows a binomial distribution ...
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1answer
31 views

Which of the following when performing a statistical test is NOT a Random Variable?

For a test for my stats class, we were asked, "When performing a statistical test on a sample, which of the following is NOT a random variable?" A) The Test Statistic B) the p-value C) The power of ...
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Cointegration of random processes

If we assume that two random processes are cointegrated, do we implicitly make an assumption about the way that the two random processes are generated?
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162 views

Why is $\mathbb E(X)=\sum_{i=1}^{n}x_i P(x_i)$?

If $X$ is a random variable and $x$'s are the realizations form $X$ and $N$ is the population size $n$ is the sample size Which one is correct $\mathbb E(X)=\sum_{i=1}^{N}x_i P(x_i)$ or ...
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24 views

Random Walk Probability Including Drift

What is the equation for the probability of a random walk with drift being equal to a specific value after n steps, given a specific standard deviation?
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164 views

What is a multivariate random variable?

I've been trying to read the Wikipedia article on multivariate random variables but I'm having trouble getting past the math. Is there a more intuitive explanation? I'm assuming that a univariate ...
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1answer
32 views

How verify if a dataset is IID

I have a dataset of 100000 samples. The samples represent the failure time of an electronic component that fails after a given number of "shocks", whatever shock means. We know that these systems fail ...
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111 views

Finding the support of transformations of random variables

Let $X, Y \sim iid U(0,1)$ and $c_1, c_2 \in \mathbb{R}$. In the linear combination $Z = c_1X+c_2Y$, we know that the probability density function of $Z$ depends on the relationships of $c_1$ and ...
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1answer
53 views

Simulating Random Vector X in R to Confirm with Example that cov(AX) = A cov(X) A'

As has been answered previously, the proof of cov(AX) = A cov(X) A' with X being a random vector and ...
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Confidence interval syntax in frequentist probability [duplicate]

Let $\theta$ be an unknown population characteristic (say average height). A confidence interval written as $P(\hat \theta - \delta < \theta < \hat \theta + \delta) = 1 - \alpha$ makes perfect ...
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2answers
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Check my proof regarding convergence in probability

I got a bit confused during the end of this proof so I am asking for a check. Take $$Y(n) = \begin{cases} 1 &\mbox{with probability} \ 1 -p_n \\ n & \mbox{with probability} \ p_n \end{cases} ...
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Can you show that $\bar{X}$ is a consistent estimator for $\lambda$ using Tchebysheff's inequality?

This question was taken from a practice exam in my statistics course. Given a random sample $X_1, X_2, ... X_n$ from a Poisson distribution with mean $\lambda$, can you show that $\bar{X}$ is ...
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random variables and dynamical systems

Dynamical systems have two parts: the state (usually a vector), and the rule (usually a matrix) such that the vector and matrix are compatible. Often enough I have seen how dynamical systems are ...
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importance of independence among random variables

I always read random variables as being independent and identically distributed. I understand the concept of being identically distributed, because if different random variables are distributed in ...
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Intuitively, why should a random variable have a distribution?

When we think about random variables or random processes, why do we make the a priori assumption that a particular realization had to come from a distribution? Why do we even have the concept of ...
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114 views

Relation between sum of Gaussian RVs and Gaussian Mixture

I know that a sum of Gaussians is Gaussian. So, how is a mixture of Gaussians different? I mean, a mixture of Gaussians is just a sum of Gaussians (where each Gaussian is multiplied by the respective ...
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1answer
38 views

Degenerate distribution

If $X \, \sim \, \mathcal{N}(m,\sigma^{2})$, I know that $\displaystyle \begin{bmatrix} X \\ X \end{bmatrix}$ is not a Gaussian vector since its entries are not independent. However, what can we say ...
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Why are observations from a random variable considered as random sample?

In a couple of books I've read a random sample is defined as a set of $n$ independent identically distributed random variables. And then their behavior is developed based on this definition which I ...
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1answer
26 views

How to obtain variance of a random variable that depends on a hypergeometric variable?

I have been given the following problem: In an assembly line production of industrial robots, gearbox assemblies can be installed in two minute each if holes have been properly drilled in the ...
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Strategy for geometric die guessing game

The first day of statistics class, we played a betting game to visualize the basics of probability distributions. It worked like this: The teacher begins by rolling a die repeatedly until the number ...
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Generate a random variable with a defined correlation to multiple existing variables

this question is strongly related to: Generate a random variable with a defined correlation to an existing variable. However I'm struggling to implement it in a more complex matter: Given X, a ...
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Generating random number based on random samples [duplicate]

I have a bunch of samples ~500 from a continuous distribution. The objective is to generate new random samples from this distribution. How to approach this problem? Can someone give me a pointer. I am ...
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Regarding convergence in probability

Let $\{X_n\}_{n\geq 1}$ be a sequence of random variables s.t $X_n \to a$ in probability, where $a>0$ is a fixed constant. I'm trying to show the following: $$\sqrt{X_n} \to \sqrt{a}$$ and ...
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PDF of dependent variables

In my recent question an answer was given, and I am able to compute it myself. Still, I'd like to understand where does that answer come from. Hence, what's the approach to handle dependent variables ...
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Compute a PDF in Mathematica/mathStatica [closed]

Let $X,Y$ be iid uniform in $[0,1]$ RVs, and $U$ has a PDF $f_U(u)=\frac{1}{4}\ln\left(\frac{4}{u}\right)$, $u\in(0,4]$. Mathematica itself is able to compute the PDF of $X+Y+\sqrt{(X-Y)^2+U}$ (see my ...
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Multivariate distribution for products of random variables

Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would ...
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Decomposing a random variable with random mean into a sum

I have two random variables: $X\sim \mathcal{N}(0,\sigma^2+1)$. $Z$, a gaussian with mean $X$, distributed so that $E_{X,Z}[(X-Z)^2]=s^2.$ We know that: $$s^2\geq\sigma^2+1 \Leftrightarrow Z ...
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PDF of a sum of dependent variables

This is a direct continuation of my recent question. The thing that I actually want to get is the distribution of $a+d+\sqrt{(a-d)^2+4bc}$, where $a,b,c,d$ are uniform in $[0,1]$. Now, the ...
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2answers
112 views

How to compute the PDF of a sum of bernoulli and normal variables analytically?

Can convolution be applied to get a closed form expression for $Z = X + N$ where $X$ is a Bernoulli random variable and $N$ is a zero mean normal random variable independent of $X$?
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What's the distribution of $(a-d)^2+4bc$, where $a,b,c,d$ are uniform distributions?

I have four independent uniformly distributed variables $a,b,c,d$, each in $[0,1]$. I want to calculate the distribution of $(a-d)^2+4bc$. I computed the distribution of $u_2=4bc$ to be ...
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Resource request : How to prove the output of a process is random variables?

I am reading through articles which present the spectral properties of chaotic systems such that they can be candidates for generating pseudo random binary sequences. One such article, is ...
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1answer
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What does it mean if the median or average of sums is greater than sum of those of addends?

I'm analyzing the distribution of network latency. The median upload time (U) is 0.5s. The median download (D) time is 2s. However, the median total time (for each data point, T = U + D) is 4s. What ...
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Distribution of the quotient of two gamma random variables with different rate parameters?

I have a question about how to derive the distribution of the quotient of two random gamma variables drawn from two different Gamma distributions with the same shape, but different rates. For example, ...
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Is a random variable Bernoulli? Is a proof available?

Suppose a die is tossed twelve times and each outcome is represented by a random variable $X_{i}$. Further define $Y_{i}$ for $i=2,...,12$ to take the value $1$ if $X_i=X_{i-1}$ and $0$ otherwise. ...
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If $X_1,X_2$ are independent beta then show $\sqrt{X_1X_2}$ is also beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities ...
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1answer
21 views

Correlation of one random variable and a binary variable dependent on it

I create one normal random variable using =NORM.S.INV(RAND()) in Excel and a binary variable =IF(B2>=0,1,0), where B2 is this random variable. If I continously resample, I find that their correlation ...
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1answer
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Confusion about the sample distribution.. Can you please enlighten me?

I thought that the sample distribution was an approximation of the distribution of the underlying phenomenon. But then the book says: We will denote the sample size by $n$ ($n \le N$) and the ...