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Course Queries Syllabus Queries 2 years ago
Posted on 16 Aug 2022, this text provides information on Syllabus Queries related to Course Queries. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution:
In my syllabus (which is in dutch, so the terms i use might be slightly off), when talking about samples, it says that
If we want to study the properties of a random variable XX in a target population, we take a random aselect sample of nn subjects from a collection of nn random variables X1,...XnX1,...Xn that are all mutually (pairwise?) independent and which all have the same distribution, namely that of XX in the target population.
A bit further, discussing convergence, it says
An infinite row of random variables X1,X2,...X1,X2,... on a probability space converges in probability to XX if the folowing is true for each ϵ>0ϵ>0: limn→∞P(|Xn−X|)≥0limn→∞P(|Xn−X|)≥0
What I don't understand is what XiXi actually means in these two contexts. I read it as follows: In the first part, it is presented as one choice from the population: XiXi is the length of the REPLY 0 views 0 likes 0 shares Facebook Twitter Linked In WhatsApp
The second paragraph simply says that if limn→∞P(|Xn−X|≥ϵ)→0limn→∞P(|Xn−X|≥ϵ)→0 then we say Xn→XXn→X in probability. That's just a definition of what "convergence in probability means".
The first paragraph talks about specific sequences XiXi - namely of those which are independent and identically distributed. You may view those as samples drawn (with replacement) from a fixed population with cumulative distribution function (CDF) DD.
You are correct that such a sequence of samples in general won't converge in probability. In fact, they never do unless the distribution of the XiXi is degenerate, i.e. there's an xx with P(Xi=x)=1P(Xi=x)=1.
They do, however converge in distribution, which is a weaker form of convergence, and means that the cumulative distribution function (CDF) of XnXn converges pointwise to DD, i.e. that limn→∞P(Xn≤x)= REPLY 0 views 0 likes 0 shares Facebook Twitter Linked In WhatsApp
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