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In the derivation of unbiased sample variance, it is considered that $X_i$ are iid random variables while $X_i$ actually represents a sample from a population. So my question is that shouldn't we consider $X_i$ to be an instance of same random variable whose probability distribution is probability of selecting $X_i$ ?

For example, consider I have population of people with different heights and I select a sample of people from this population. This sampling of people can be thought of a repetitive procedure of sampling a value from a random variable i.e uniform r.v. Isn't it so?

Also I have generally seen it that when we consider samples of a stochastic process, we model each sample as i.i.d instead of repetitive sampling (taking multiple values) from the stochastic process. For example in the definition of strict sense stochastic process we say that the joint distribution of different samples of $X_t$ will be same whether we sample it at any time. Here my question will be rephrased as how can we have joint distribution of constant numbers? Like if I have a series of dice-face-numbers then there is no meaning of considering joint probability distribution of these numbers?

user_3pij
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  • The first two question you pose... I've asked myself the same questions multiple times. I think I even posted some sort of question here but I am not sure about that. Treating these $X_i$ as different random variables even though i.i.d. ones... it's just somehow counter intuitive to me. – peter.petrov Jan 20 '21 at 21:23
  • But also... I am pretty sure that there's no such thing as "different instance of same random variable" – peter.petrov Jan 20 '21 at 21:27
  • Here are several related questions: Q1, Q2. Q3. –  Jan 20 '21 at 21:35
  • Thanks @d.k.o. I think following threads cleared my understanding. https://math.stackexchange.com/questions/1760048/what-is-a-sample-of-a-random-variable https://stats.stackexchange.com/questions/141416/example-of-sample-x-1-x-2-ldots-x-n?rq=1 https://math.stackexchange.com/questions/3324984/terminology-regarding-random-sample https://math.stackexchange.com/questions/2084035/confused-about-the-definition-of-a-random-sample-statistics-and-estimators-esti?rq=1 https://math.stackexchange.com/questions/3429350/difference-between-mean-for-samples-and-mean-for-random-variable?noredirect=1&lq=1 – user_3pij Jan 21 '21 at 01:13
  • In short, the word "sample" is used differently depending upon the context. Sometimes it's is used as a realization of a random variable (which I was thinking). But most of the times a sample of size n of a r.v $X$ is actually a collection of n independent r.v $X_i$ which have identical distribution as that of $X$. The context depends on your experiment. It usually means collection of values(realization) in a survey context where you collect a survey i.e you are actually recording values and means a collection random variables where you randomly select samples from a population. (1/3) – user_3pij Jan 21 '21 at 01:51
  • For example if an experiment is such that I toss a coin 2 times (or tossed two coins simultaneously) and record values according to face of coin then the recorded values will be a sample(member) from the population {00,11,01,10}. Now If I say that I tossed a coin two times (or tossed two coins simultaneously) then a sample can be represented as $X_1$, $X_2$ where $X_i$ is a Bernoulli random variable. (2/3) – user_3pij Jan 21 '21 at 01:51
  • If a say that I RANDOMLY select (or observe) an element of population n times where the probability distribution of selecting the element is given by random variable $X$ then it always means that a sample is collection of n iid random variables. Hope it helps! (3/3) – user_3pij Jan 21 '21 at 01:51
  • Such questions typically get better answers over at cross validated. Here are some posts there to consider: https://stats.stackexchange.com/questions/440519/differences-between-realization-of-the-random-variable-and-deterministic-variabl (and a lot of similar). Maybe it is better to go to the conceptual foundation in the concept of exchangeable see https://stats.stackexchange.com/questions/344794/exchangeability-and-iid-random-variables, https://stats.stackexchange.com/questions/233259/why-are-words-in-a-document-for-bag-of-words-model-exchangeable-but-not-independ, – kjetil b halvorsen Jan 22 '21 at 17:16
  • ... https://stats.stackexchange.com/questions/207162/simple-question-on-exchangeability, https://stats.stackexchange.com/questions/3520/can-someone-explain-the-concept-of-exchangeability, https://stats.stackexchange.com/questions/447383/is-independence-subjective and then continue search on CV (or maybe here?) for post on exchangeability (and the deFinetti Theorem.) – kjetil b halvorsen Jan 22 '21 at 17:20

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