Questions tagged [iid]

iid is an acronym for independent and identically distributed. Many statistical methods assume that the data are iid; that is, that each observation comes from the same distribution and is independent of other observations.

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Is the Distribution of Survival Times always IID?

I was reading about an approach to Survival Analysis called "First Hitting Time Models" (threshold regression): https://www.jstatsoft.org/article/view/v066i08 , Can Survival Models model the ...
Uk rain troll's user avatar
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36 views

Assumptions about data distribution for OLS [duplicate]

From all the content I have read, ordinary least squares (OLS) assumes that the errors are normally distributed i.i.d. I still cannot find any content that gives a bit more insight on what can be ...
spie227's user avatar
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Misunderstanding on the use of Popoviciu and von Szokefalvi Nagy's inequalities on the variance of a unbiased estimator

Let $X_1,\cdots,X_n$ be (discrete in my case) i.i.d. and bounded between $m$ and $M$. I'm interested in bounding the variance of an unbiased estimator: $$\mathbb{V}\left[\frac1n\sum_{i=1}^nX_i\right]$$...
Tristan Nemoz's user avatar
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If $X_1, \dots, X_n$ iid, are $f(X_1), \dots, f(X_n)$, also iid? [duplicate]

If I have independent and identically distributed random variables $X_1, \dots, X_n$, then are $f(X_1), \dots, f(X_n)$ themselves independent and identically distributed? I think the answer is yes, ...
caitlin's user avatar
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Is a random walk i.i.d.? [closed]

Assuming that we have an AR(1) process: $$X_t=\rho X_{t-1}+\varepsilon_t,\quad\varepsilon\sim IIDN(0,1)$$ And further assuming that $\rho=1$ and $X_0=0$ we have a random walk process: $$X_t=\sum_{i=1}^...
Rstrobaek's user avatar
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97 views

How to find the MGF of the max of a set of i.i.d. exponential random variables

As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
stats6895997's user avatar
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Is a random subset of exchangable data still exchangable? How about iid?

I am dividing the data that I work with (iid, thus also exchangable) into training and calibration sets based on the random sampling (drawn from Bernoulli dist with parameter q). Does this still ...
Anđela Todorović's user avatar
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48 views

About the requirement of iids to form a martingale

Most problems about martingales start with an assertion similar to the following one: Suppose that $X_n$ are iid such that $\mathbb{E}[|X_n|]<\infty$ and $\mathbb{E}[X_n]=0$. Then $S_n = \sum_i^n ...
user3141592's user avatar
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Are these trials identically distributed? These trials from a panel data are all Bernoulli experiments; do they have same probability of success p?

Scenario: I have an ordered/indexed sample that supposedly comes from a binomial distribution with fixed probability of success p. (By ordered/indexed I mean that not only I care about the percentage ...
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iid condition of SPC

The control chart usually requires that the observed values meet the conditions of independent and same distribution, but the sample data is a time series with autocorrelation. How should the data be ...
xin liu's user avatar
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IID assumption in proportion hyp test

I am asked to test a hypothesis that a manufacturing line makes p% faulty parts in a month, it's assumed that the p% is independent of the month. My approach is as simple as it gets, take a random ...
abdelrahman esmat's user avatar
1 vote
2 answers
50 views

Can i.i.d. random variables be negative? [closed]

I know that random variables can take negative values, so can independent and identically distributed random variables be negative?
Cathy's user avatar
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How to test whether a given set of data possesses serial dependence or not?

I have a dataset where there are measurements taken over time. I have no assumptions whatsoever regarding the exact nature of the process, i.e. whether the measurements are iid or belong to a time ...
Ghostpunk's user avatar
3 votes
1 answer
216 views

If a strictly stationary process is also independent, does this imply i.i.d.?

Suppose I have a time series process $\{X_t\}$ that is strictly stationary in the sense that the joint distribution of $[X_{t_1},...,X_{t_k}]$ and $[X_{t_1+a},...,X_{t_k+a}]$ are the same for any set ...
ExcitedSnail's user avatar
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2 votes
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Statistical power of an experiment calculation

As titled,I had some confusion related to statistical power calculation. The calculation is done using ...
Alex's user avatar
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112 views

Proving that a series is white noise, but not IID

I have this problem that I'm struggling to solve Be $\left\{ Z_t \right\} \sim N(0,1)$ a stochastic process and let: $$X_t = \begin{cases} Z_t & \text{if $t$ is even }\\ \left( ...
Tricolor's user avatar
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1 vote
1 answer
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Permutations of iid Random Variables [duplicate]

Suppose that $X_1, X_2, X_3$ are iid random variables. I have seen this fact many times that $$\mathbb{P}(X_1<X_2<X_3)=\frac{1}{6}$$ but I want to know that why every permutation of $X_1, X_2, ...
user671269's user avatar
1 vote
2 answers
220 views

Variance of the difference of two iid sample means

Let $X_{1}, ..., X_{n}$ be random variables independent of $Y_{1}, ..., Y_{n}$, where both groups are iid with associated population means $\mu_{1}$ and $\mu_{2}$ and population variances $\sigma_{1}^{...
Stanley Yu's user avatar
1 vote
1 answer
195 views

How to: Bootstrap Prediction Intervals for Regression Models with non IID noise?

Question: Consider a regression model $Y=m(X)+\epsilon$ for which $\epsilon$ is neither independent of $x$ nor identically distributed. How would we go about generating prediction intervals in such a ...
TheFriendlyAsker's user avatar
-2 votes
1 answer
42 views

Is part of the sequence of independent identically distributed random variables still iid? [closed]

if $x_0,x_1,...x_n$ be iid (independenta and identically distributed) random variables. The question is that the $x_0,x_3,...x_{n-1}$ is also iid?
Cathy's user avatar
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2 votes
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37 views

A constant $c$ minus the iid random variables still iid? [duplicate]

Let $x_0,x_1,...x_n$ be iid (independenta and identically distributed) random variables. Then, $m_0,m_1,...m_n$ be defined as $c-x_0,c-x_1,....c-x_n$, where c is a constant greater than $x_i$ ($i \in \...
Cathy's user avatar
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74 views

Empirical distribution function by sampling from a m.v. distribution

I have mathematically rewritten my problem as a function of multiple iid variables: $$ f(X_1, X_2, ..., X_n), $$ where $$X_i \in \mathcal{N}(0,1)$$ I now want to determine the empirical distribution ...
Filip Johansson's user avatar
3 votes
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40 views

Are there examples of ML or stats approaches that are valid for IID data, but not exchangeable data?

Lots of supervised learning theory is motivated using the IID assumption. Do most of these methods apply equally well if data is only exchangeable, and not IID? Can you provide an example where this ...
capybaralet's user avatar
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388 views

What is the CDF of sum of 2 Pareto random variables?

Assume we have two independent, random variables, $X$ and $Y$ which follow the same Pareto distribution, i.e.: $f(x) = \frac{1}{x^{2}}, x > 1$ Given that - what would be the joint probability $P(X +...
macgivera's user avatar
1 vote
1 answer
251 views

Central limit theorem for asymptotically i.i.d. random variables

I observe a sequence of r.v. $X_1, X_2, \dots$ where each $X_i$ is a function of the sample size $n$. When $n \rightarrow \infty$ I have the following result: $X_1 \rightarrow^d E_1, X_2 \rightarrow^d ...
Eryna's user avatar
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2 votes
2 answers
83 views

If $X_1,\ldots,X_n$ are continuous and IID with sample median $m$, are $1(X_1\geq m),\ldots,1(X_n\geq m)$ IID?

Suppose $X_1,\ldots,X_n$ are continuous random variables and IID with sample median $m$ for some $n\geq 2$. Is it then true that $1(X_1\geq m),\ldots,1(X_n\geq m)$ are IID? Here the definition of $1(...
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Data collection after the model is built and deployed

I have built a machine learning model which predicts whether a customer will buy a product or not. The model performs well on cross validation tests. Now, I will deploy it in production to recommend ...
Sanyo Mn's user avatar
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2 votes
1 answer
362 views

Does the assumption of “independent and identically distributed” apply to the actual sample data or the sampling process?

While studying statistics, I came across the concept of “independent and identically distributed random variables” or IID. I’m confused as to what that applies to in practice, mainly, does it apply to ...
user216096's user avatar
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26 views

$E(SN)$ for aggregate claim amount $S$, $S=X_{1}+...+X_{n}, X_{i}$ are iid [duplicate]

Consider the following model for aggregate claim amounts $S$: $S=X_{1}+X_{2}+...+X_{N}$ where the $X_{i}$ are independent, identically distributed random variables representing individual claim ...
Pallak Goyal's user avatar
2 votes
1 answer
71 views

Studying extreme value r.v. $X=\max_i (c_i+X_i)$ where $c_i$ are constants and $X_i$ are i.i.d. r.v

Let $X_1,X_2,...,X_n$ be independently and identically distributed random variables according to a distribution $F$. There are constants: $c_1,c_2,...,c_n$. Define a new random variable $X=\max_i(X_i+...
dodo's user avatar
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2 votes
0 answers
91 views

Test for distribution equality

This question touches Kolmogorov-Smirnov testing, but asks actually something different. Consider independent random variables $X_1, \dots, X_n$. I want to test the following hypothesis: $$ H_0: X_i\ ...
Albert Paradek's user avatar
2 votes
1 answer
316 views

MCMC Sample should be i.i.d

I'm a bit not sure how to show that MCMC samples are i.i.d. In my opinion the trace plot should behave like white noise model because white noise model has a strong stationary properties i.e. the ...
actsci stud tries2learn math's user avatar
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393 views

If we remove half the samples from an IID dataset, is the remaining half still IID?

I am generating 10,000 pairs of X and Y such that both X and ...
Kristada673's user avatar
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0 answers
59 views

understanding "independent" term in "independent and identically distributed" (iid) statement

I understand independent as the random variable values in a data aren't connected to each other in any way. (q1) Is my understanding ok? data_1: a product which has a design defect has been sold to ...
Ersin's user avatar
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6 votes
1 answer
287 views

Bernoulli distribution with random means

Let $S = \frac{X_1 + \cdots + X_n}{n}$ where the $X_i$ are IID Bernoulli distributed with mean $p$, then $E[S] = p$ and $Var(S) = \frac{p(1-p)}{n}$. Now consider the slightly more complex setup where $...
gauss's user avatar
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1 answer
575 views

Properties of independent and identically distributed random variables [closed]

Do independent and identically distributed random variables always have the same expectation and variance?
NIkhil 's user avatar
9 votes
1 answer
415 views

References on data partitioning (cross-validation, train/val/test set construction) when data are non-IID

Consider a prediction setting in which we are interested in training a regression or classification function $f$ with inputs $X \in \mathbb{R}^k$ and target $Y$, and assessing its expected ...
Adrian's user avatar
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10 votes
3 answers
2k views

Statistical learning when observations are not iid

As far as I am concerned, statistical/machine learning algorithms always suppose that data are independent and identically distributed ($iid$). My question is: what can we do when this assumption is ...
riccardo-df's user avatar
2 votes
2 answers
184 views

Confusion about independent and identically distributed?

Say that I wish to measure the height of male within the population (so gender=Male is the only factor I am accounting for). Say I collect 100 observations of male height from an elderly population. ...
Michael's user avatar
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3 votes
2 answers
206 views

Asymptotic MLE Distribution With Two Random Samples

I'm studiyng for an exam, and I found this problem which I can not managed to solve... I will be really grateful if someone can help me, thanks you. Let $\left\{X_{1}, \ldots, X_{n}\right\} \sim^{...
Cornflake's user avatar
1 vote
0 answers
176 views

Central limit theorem for dependent binary-related variable

Let $Y\sim N(\mu, \sigma^2)$ and given sample size $n$, we have an iid sample $\{Y_1, ..., Y_n\}$. We sample $X$ (size $n$) from Bernoulli with probability $\pi$. Denote $Z_i=X_iY_i$. Then, when $X_i=...
TrungDung's user avatar
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2 votes
1 answer
501 views

Types of noise processes and the one assumed in arima() estimation in R

Here is a time series class defining white noise incorrectly as an independent sequence of random variables. source Aside from the widespread mix-up of White noise and iid noise, a further ...
ColorStatistics's user avatar
4 votes
1 answer
7k views

What the i.i.d. assumption of the errors in linear regression implies for the response variable y?

In the linear regression model we assume that the errors $ε_i$ are independent and identically distributed (i.i.d.) random variables. I am trying to understand what this assumption implies regarding ...
gnikol's user avatar
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1 vote
1 answer
363 views

AR/ARMA model - test residual for independence or lack of correlation

Shumway and Stoffer (2017), a great book that I highly recommend, define an AR(p) model as $x_t=\phi_1 x_{t-1}+\phi_2 x_{t-2}+...+\phi_p x_{t-p}+w_t$, where $w_t$~$wn(0,\sigma_w^2)$; $w_t$ is not ...
ColorStatistics's user avatar
1 vote
0 answers
48 views

What is the meaning of these subscripts in iid random variables? [duplicate]

I've been studying statistics on my own and I'm having a hard time understanding some notations. On this page: http://scipp.ucsc.edu/~haber/ph116C/iid.pdf, specifically on the second paragraph, the ...
RafaelSantiago94's user avatar
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0 answers
274 views

MGF of sample mean of poisson distribution

Let $X_1,X_2,\dots,X_n\stackrel{iid}{\sim}Poiss(\lambda)$ Let mgf of $X_1$ is given by $M_X(t)=e^{\lambda(e^t-1)}$ and let $\bar{X_n}=\frac{1}{n}(X_1+X_2+\dots+X_n)$ Then, by Weak Law of Large Numbers ...
abhishek's user avatar
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0 votes
1 answer
371 views

Clarifications on I.I.D. assumption in machine learning

In this question, it was stated that the assumption of i.i.d. for data comes in the form of $$(X_i,y_i)∼P(X,y),∀i=1,...,N \\(X_i,y_i) \;independent\; of \;(X_j,y_j),\;∀i≠j∈{1,...,N} $$ I am clear with ...
tangolin's user avatar
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0 votes
1 answer
111 views

Confusion in definition of independent and identically distributed random variables [duplicate]

From what I learnt, a random variable is a function which assigns real values to outcome space, and the probability distribution is a function that assigns probability to different values produced by ...
newbLearner's user avatar
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0 answers
54 views

Obtaining better estimates when you know that a set of input variables are independently and identically distributed

Suppose I have a family of random variables $$X_i \sim SomeDistribution_i, \ \ i = 1,..., n$$ and I know how to sample $SomeDistribution_i$ for any $i$. Suppose I also define a random variable $Y = f(...
user118967's user avatar
2 votes
0 answers
51 views

ISI PCB NC$9$ Limiting Distribution of Bernoulli to Poisson

Let $X_i\sim (i.i.d.)$, Bernoulli($\frac{\lambda}{n}$), $n\ge \lambda\ge 0$. $Y_i\sim (i.i.d.)$, Poisson($\frac{\lambda}{n}$). $\{X_i\}$ and $\{Y_i\}$ are independent. Define $T_n=\sum_{i=1}^{n^2}X_i$...
Kumar's user avatar
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