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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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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12 views

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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3answers
37 views

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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2answers
40 views

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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2answers
81 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
35 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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26 views

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
22 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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0answers
38 views

Question about the departure process of a M/GI/$\infty$ queue

Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...
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2answers
36 views

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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1answer
19 views

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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0answers
30 views

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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118 views

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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1answer
199 views

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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0answers
33 views

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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0answers
17 views

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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48 views

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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91 views

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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99 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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212 views

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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2answers
24 views

Resource request : Relationship between chaotic dynamics and $i.i.d$ 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
44 views

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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1answer
97 views

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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2answers
127 views

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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264 views

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
18 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
50 views

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 ...
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2answers
97 views

Random variables independence

I need to check if $Z$ and $W$ are dependent or not. $X,Y \sim \mathrm{Exp}(2)$ Then I define: $Z=X-Y \ \text{,}\ \ W=X+Y$. Now How I can check that $Z$ and $W$ are dependent or not ? I know ...
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1answer
63 views

Example of sample $X_1,X_2,\ldots,X_n$

In the book Statistical Inference by George Casella, it is written that An experimenter uses the information in a sample $X_1,X_2,\ldots,X_n$ to make inferences about an unknown parameter $\theta$. ...
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1answer
35 views

Chi-squared distribution and dependence

We know that for a group of independent random variables $\{X_i\}_{i=1}^n$ s.t each $X_i$ is distributed as a chi-squared distribution with degree of freedom one ($\chi_1^2$), the sum of these random ...
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15 views

Why is it inappropriate to use a post hoc test on a random effect?

This is a bit more conceptual, but I know its incorrect to run a post hoc test on a random effect, though I'm not entirely sure as to why. I figure it has to do with the selection of random variables, ...
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39 views

Finding MLE with ordered statistics?

Let Y1 < Y2 < ... < Yn be the order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval: $$[\theta - \rho, \theta + \rho]$$ ...
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36 views

CDF of the function of a random variable

I haven't been able to find useful information on this. I was just wondering what would be the distribution of a function of a random variable. For example, what would be the distribution of the ...
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3answers
119 views

What is the difference between variable and random variable?

I know that "variable" means "values which vary." In a simple linear regression model : $$Y=\beta_0+\beta_1X+\epsilon$$ $X$ is variable that is the values of $X$ vary. Why is $X$ not a random ...
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116 views

Probability of finding a point in the unit circle?

Consider the experiment where a pair of numbers (x,y) is chosen at random in the unit square; that is, x and y are uniform (0,1) random variables. What is the probability of (x,y) lying within the ...
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29 views

Is sampling according to several i.i.d. random variables or to just one of them equivalent

Suppose we have n+1 random variables i.i.d. distributed $X_0,X_1,...X_{n}$. Is it the same operation if I generate n samples according to $X_0$ $\{S^0_1,S^0_2..,S^0_n\}$, or if I take from each of the ...
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16 views

How to find anti-correlated subsequences in correlated time series?

Say I have two time series $X_t$ and $Y_t$ (with $ 1 \leq t \leq N$), which have a high positive Pearson correlation. Say I also have reason to believe there are subsequences $X_{tj}, Y_{tj}$ (where, ...
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97 views

Parameters vs latent variables

I have asked about this before and have really been struggling with identifying what makes a model parameter and what makes it a latent variable. So looking at various threads on this topic on this ...
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1answer
59 views

PDF of function of X

I'm learning about functions of random variables and am trying to work out an example I made up. If $y = \sin(x)$ and $x$ has domain $[0, 4\pi]$, is the following the correct expression for the pdf of ...
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1answer
43 views

On an implication of the memoryless property of the exponential random variable

I know that if we take $X \sim Exp(k)$ then we have this property: $$P(X \ge s + t | X \ge s) = P(X \ge t)$$ But why does this imply that $X | X > x$ has the same distribution of $X$ only ...
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32 views

“negligible” random variables

I would like to ask if the concept of 'negligible' random variables exist. For example, if $X$ and $Y$ are random variables, not necessarily independent, then the sum $X+Y$ is, for all intents and ...
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1answer
27 views

Variance calculation RELU function (deep learning)

weight initialization is important for modern deep learning. To understand [1,2], I would like to understand the following: $$ E[x^2] = 0.5 Var[y], $$ where $x= max(0,y)$, $E[.]$ is the expectation, ...
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1answer
45 views

Distribution of the product of a gamma random variable and a beta random variable

When you multiply a gamma random variable with a beta random variable, you should get a gamma random variable. I'm having a little trouble showing this, though. I figure I'm forgetting some clever ...
3
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1answer
113 views

correlation of r1 and r2 is x. Probability of r1 > r2?

Just a quick probability interview question. If the correlation of two variables r1, r2 is x. What's the probability that a sample of r1 is greater than a sample of r2? let me update the question. ...
3
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1answer
56 views

Continuous random variable and median problem

The definition of median, $m$, for a continuous random variable $X$ is $$P(X\leq m)=P(X\geq m)=\int_{-\infty}^m f(x)\text{d}x=\int_{m}^{\infty}f(x)\text{d}x=\frac{1}{2}$$ where $f(x)$ is the pdf of ...
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26 views

Notation for a random vector whose length depends on another random variable?

I have the following process that I'm trying to describe with random variables. First, I have a random variable $X$ that takes on values drawn from a Poisson distribution with parameter ...
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1answer
64 views

Sum of iid random variables

Let $X_1, X_2,...,X_n$ be iid random variables. Let $Z_1, Z_2, Z_3$ be defined as $X_1, X_1+X_2, X_1+X_2+X_3$ respectively. Are $Z_1, Z_2$ and $Z_3$ also iid's? The question is based on renewal ...
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3answers
62 views

Intuition of pdf of a continuous random variable [duplicate]

What is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if ...
2
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1answer
13 views

Using variables that are only available for part of the data-set in a classification model

I have Data X1, X2, and y. X1 has the same variables as X2, + some extra variables that X2 does not have. I want to use the data X2 to predict binary variable y. I suspect the extra variables In ...
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1answer
64 views

Mixed effect modelling with multiple, nested random variable

Goal: comparing pitch (Hz) on three types of words Dependent variable: Hz Fixed predictor variable: word-type, points (measurements taken from five points on each token, to capture Hz change within ...