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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14 views
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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1answer
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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0answers
41 views
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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0answers
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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0answers
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-...
2
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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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1answer
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
$$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}}
\Var(AX+BY+c)=A\Var(X)A′+ B \Var(...
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2answers
72 views
Why is the risk function defined to be the expectation of loss function?
In decision theory, we define the risk associated with a particular predictor function as the expected value of the loss function. Since the input and output are considered random variables therefore ...
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1answer
33 views
Equivalence of the probability distribution of a symmetric function = 1/2
Let $\mu \in \mathbb{R}$ and suppose the probability density function $f$ of the random variable $X$ satisfies $$f(x-\mu) = f(x+\mu) \quad \forall x \in \mathbb R.$$
Show that $F(\mu) = \frac{1}{2}$,...
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0answers
30 views
$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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0answers
20 views
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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0answers
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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0answers
33 views
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 ...
3
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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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0answers
68 views
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
109 views
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 ...
2
votes
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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0answers
6 views
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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0answers
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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0answers
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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0answers
9 views
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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0answers
24 views
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 ...
3
votes
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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0answers
44 views
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?
2
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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 ...
2
votes
2answers
98 views
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
39 views
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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0answers
22 views
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 \...
7
votes
5answers
1k views
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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0answers
37 views
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 \...
2
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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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0answers
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 ...
2
votes
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 ...
2
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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 ...
3
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0answers
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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1answer
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 ...
