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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 calculate the average number of events per interval in poisson distribution

Hi dear statisticians, I have a random variable X that follows a poisson distribution. However, I only know the number of occurrences in an interval and the cumulative probability a. How I can ...
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1answer
55 views

The definitions of estimator and estimate

This example demonstrates the difference between a theoretical observation and a realized observation. A theoretical observation is a random variable with a probability distribution, while its ...
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32 views

Is this method of finding the expected value of the square a random variable correct?

Suppose x is a discrete random variable with values 2,3,1 and probabilities 0.2,0.3, and 0.4 respectively. NOw say we have the function y=x2+3 and we want to find the expected value of this equation. ...
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1answer
45 views

why X|Y=y isn't considered to be a random variable

I read about conditioning of random variables and I got a little bit confused. Why $X|Y=y$ isn't considered to be a random variable (it is just a function of $y$'s) while $E(X|Y=y)$ can be considered ...
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Expectation of product

Let $\{X_i\}_{i\in I}$ be a finite collection of i.i.d random variables. I have found that $$E[\prod_i X_i]=\prod_i E[X_i]$$ But I haven't found a proof of this fact. What is the proof?
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23 views

Parameter Estimation with Method of Moments

I was working on some problems and came across this one I was having trouble with: Given a random sample {0.1, 0.4, 0.2, 0.1} from a Beta population with β = 1, use the method of moments to find an ...
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9 views

Introducing a Bernouli random to ensure expecation remains the same after quantification

I'm going through a paper where they use a compressed floating point representation to save space when communicating the values over the network. Each floating point value $q$ is represented as a d-...
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1answer
25 views

Sampling from specific distribution

Suppose I have two random variables $X$ and $Y$ that are independent. Also suppose that I can sample from $X+Y$ and $Y$. Is it possible to combine those two sampling algorithms to get samples for $X$.
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9 views

A generic terminology for a given object to be estimated?

If one has to suggest, what is it one would call a given object that is to be estimated? We already have a generic term "estimator" on the one hand. When the context is clear, usually there is no ...
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1answer
47 views

Variance of sum of random vectors - a proof

For nonrandom matrices $A(rXk)$,$ B(rXm)$, and $c(rX1)$, how does one show that $$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}} \Var(AX+BY+c)=A\Var(X)A′+ B \Var(...
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Why is the risk function defined to be the expectation of loss function?

In decision theory, we define the risk associated with a particular predictor function as the expected value of the loss function. Since the input and output are considered random variables therefore ...
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1answer
33 views

Equivalence of the probability distribution of a symmetric function = 1/2

Let $\mu \in \mathbb{R}$ and suppose the probability density function $f$ of the random variable $X$ satisfies $$f(x-\mu) = f(x+\mu) \quad \forall x \in \mathbb R.$$ Show that $F(\mu) = \frac{1}{2}$,...
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$V=XY/\sqrt{(X^2+Y^2)}$ Distribution [duplicate]

If $X$ and $Y$ are iid N(0,1) what is the pdf of $V=XY/\sqrt{(X^2+Y^2)}$. I have found out the distribution of $1/(X^2)$. So will that be any useful and if yes how ?
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Random variable not associated with an event? How? [duplicate]

How do I understand/imagine/visualize a random variable not being associated with an event - for example, in the case of Bayesian statistics in contexts such as in maximum a posteriori estimation? (...
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29 views

Distribution of $X_1X_2+X_3X_4$ [duplicate]

Let $X_1,X_2,X_3,X_4$ be independent $N(0,1)$. Find the distribution of $Y= X_1X_2+X_3X_4$ I am having difficulty in solving this question or atleast how to approach it.
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3answers
133 views

$P(\lvert X - \mu\rvert \geq \sigma)$ as a measure of tailedness

I know that one of the standard measures for the "tailedness" of a distribution is kurtosis, i.e. fourth standardized central moment $\frac{\mu_4}{\sigma^4}$. This measure is sort of intuitive to me: ...
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How do I find the mean of Z = Discrete RV + a Gaussian RV?

I am asked to find mean and variance of Z. (image) Since I am solving a preparatory examen to study, it is not clear to me how to approach the topic because I don't understand the question correctly. ...
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2answers
32 views

Covariance of random variables whose sum is less than a constant

Suppose that we have integer random variables $X>0$ and $Y>0$ and constant number $a$. We have: $X+Y < a$. Can we say that the covariance of these random variables is less than or equal to ...
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1answer
30 views

Tail bound for sum of i.i.d. random variables with common moment generating function

Suppose $\{X_n\}_{n\in \mathbb{N}}$ is a sequence of independent and identically distributed random variables and $S_n:=X_1+...+X_n$. Assume that each $X_i$ has mean $0$ and that all $X_i$ have a ...
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0answers
22 views

Minimum of i.i.d. Random Variables [duplicate]

What importance does the minimum of an identical and independently distributed random variables play in probability distribution?
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1answer
36 views

How to lower the standard deviation in a Monte Carlo Simulation [closed]

I am trying to simulate a stock's price with a Monte Carlo simulation. I am using this formula in excel: $S_{t+1}=S_t\cdot exp(d\Delta{t}+s\varepsilon \sqrt{\Delta{t}})$, where $d=\bar{x}-\frac{s^2}{2}...
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Does the normal product distribution have subgaussian tail?

Consider the normal product distribution, which is the distribution of the product of two or more independent normal variables. Particulary, focus in the case where the multiplied normal variables are ...
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2answers
45 views

How to find distribution function of sum of 2 random variables that are uniformly distributed? [duplicate]

I am stuck with this tutorial question in one of my stats module and I would greatly appreciate some help: Let $X1$ and $X2$ be independent random variables with $a = 0$ and $b = 1$ i.e. $X1$ and $X2$...
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2answers
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About Sampling and Random Variables

So I've recently started an introductory course in econometrics and I'm having trouble grasping the idea of Random Variables and Sample distributions If we have a population and we take a sample $Y= \...
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1answer
105 views

Basic: Why are Slope, Intercept in Regression considered Random Variables?

Sorry if this is too basic. In an OLS regression given by $y=ax+b$ $b$ intercept, $a$ the slope. Then $a,b$ are not numbers but random variables. I find this confusing since I start with data ...
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2answers
26 views

Getting variance of function of two uniform RVs [duplicate]

Have two independent RV's $X$ and $Y$ sampled uniformly from $[0,1]$ and $C = (X-Y)^2$. Want $V(C$). Rewrote as $V((X-Y)^2) = V(X^2) - 4V(X)V(Y) + V(Y^2)$ but that's too messy. Is it correct to write ...
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Formulating model to measure the recency (in time) of a random variable

I run a social news ranking web application (built in Python) where users post items and vote on others' such postings. I am trying to curtail Sybil nodes so that disingenuous voting can be rooted ...
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25 views

Expectation of random sum of dependant variables

The expectation of random sum of independent identically distributed variables is given either by the law of total expectation or by Wald's identity. Are these generalised to tackle the random sum of ...
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26 views

Implicitly defined random variable

Recently I came across the following problem: Let $f:\mathbb{R}^2\to\mathbb{R}$ be a function as smooth as we want (even analytic if this is convinient) such that the equation $f(x,y)=0$ for every $x$ ...
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Unit Step Functions

The image below shows the process for calculating the Expected value of a squared random variable X. From line 3 to 4 the unit step function [u(x) - u(x-1)] is removed and a 3 is placed before (1-x^2)....
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sub-exponential property of a linear transformation of independent random variables

I need to use a Bernstein-type bound to a random variable $y^Ty$, where $y= Qx$. $x$ is a Gaussian vector where each entry is independent. $Q$ is a kernel matrix. So $y$ is the linear transformation ...
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1answer
66 views

Correct understanding of De Finetti`s representation theorem

I am currently interestend in understanding De Finetti`s representation theorem. As I am only familiar with Frequentist thinking I have some problems to understand its meaning. I have already read the ...
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0answers
51 views

Showing that two random variables are independent [closed]

If $f(x,y) = g(x+y)$ with $0 \leq x,y \leq 1$ and $0 \leq x+y \leq 1$ then are $X$ and $Y$ independent?
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1answer
22 views

Covariance between a variable and a non-linear transformation of it

Suppose $\epsilon \overset{\text{iid}}{\sim} N(0, \sigma^2)$ Can we make any assumptions about Cov$(\epsilon, \frac{\epsilon^2}{1 + \epsilon^2})$?
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1answer
33 views

Geometric interpretation of mathematical expectation of a random variable

Is there any nice geometric interpretation of the mathematical expectation of a random variable (preferably based on density or cumulative density plot)? (For example, median has a nice geometric ...
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Can i say an iid process with $0$ mean has homogeneous independent increment?

I think the title is itself self-explanatory. My idea says that an IID process with $0$ mean has to have independent increment. But, I do not understand how to prove homogeneity?
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1answer
46 views

Expected value conditional on a function

Let $X$ and $Y$ be random variables. What is the relationship (if any) between $E[Y|X]$ and $E[Y|g(X)]$? I have been trying to Google or look in books but I'm having trouble even articulating this ...
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2answers
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Suppose $y=h(u)$ prove$E(Y)=E(h(u))$ [closed]

Suppose $y$ has a pdf $f(y)$, and $y=h(u)$ for some variable $u$, where $g(u)$ is the pdf for $u$. Prove that $E(Y)=E(h(U))$ that was to prove $\int yf(y)dy=\int h(u) g(u)du$.
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1answer
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How do you check that a sampler and a density correspond to the same random variate?

General Question If someone handed you a direct sampling algorithm and a density function, and they told you that the two corresponded to the same random variate, how would you check this? ...
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Question on Law of Large numbers argument including conditioning

Consider a random process $X_n$ that may be non i.i.d. Let $\epsilon >0$. Assume that for $X_n$ the following equality holds \begin{equation} \sup_{\ell \geq 1}\text{ess} \sup \lim_{n\rightarrow \...
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5answers
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Random variable defined as A with 50% chance and B with 50% chance

Note: this is a homework problem so please don't give me the whole answer! I have two variables, A and B, with normal distributions (means and variances are known). Suppose C is defined as A with 50% ...
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1answer
50 views

Expectation of random variables

Suppose, $X_1,X_2,X_3,X_4$ are i.i.d. random variables with values 1 and -1 with prob 0.5 each. Then find the value of $E(X_1+X_2+X_3+X_4)^4$. Ans: Let, $Y=\sum_{1}^{4}X_{i}$ Now, Y takes values 4,-...
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Proof: sum of two normally distributed random variables [duplicate]

I'm trying to wrap my head around the sum of two normally distributed random variables. If I have the following: $$\begin{array}{l} X \sim \mathcal{N}\left( {{\mu _X},\sigma _X^2} \right)\\ Y \sim \...
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0answers
35 views

Create a random dataset for regression analyses (teaching and model testing) [closed]

I would like to create a random dataset which allows me to conduct regression analyses as realistically as possible (e.g. for teaching, but also to test different models). The analyses I would like to ...
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1answer
20 views

Calculating a mixed model using pdIdent var-cov matrix

I have a question in mixed models. I'm quite new at this field and I'm trying to calculate a model in which "y" is predicted according to multiple covariates (Age,gender,BMI) and the random variable "...
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23 views

Expressing binomial distribution

Consider the following random variables: $$ Y := \sum_{i = 1 }^{n-1} X _i \quad Z:= \sum^n_{i=1} X_i $$ Further we introduce a new variable $k \in \mathbb{N} $ $1 \leq k \leq n-1 $, meaning $k \neq ...
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1answer
130 views

Mean of a censored Poisson random variable

Consider a real demand estimation problem of a retailer where matrix $Y$ (contains Poisson random variables) is the real demand and its mean need to be estimated by using sales data (matrix) $X$ which ...
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0answers
21 views

Independent variables in truncated Weibull distribution

I am relative new to statistics and currently working on a problem about crash rate analysis. It is appears to be a zero-inflated scenario, then I decide to use the hurdle model. The first part will ...
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51 views

Taylor Series Expansion of Unconditional Expectation

We know that the best 1st order approximation of an unconditional expectation is the following- $$E(y|x)=(E(y)-\beta E(x))+\beta x$$ where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}...
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33 views

Why does the PDF use a different variable than x?

In the below image (from Wikipedia but also found in my text book), I noticed that the variable within the integrand is a "u" rather than the "x" which is found in the CDF function. Why is the ...