Thank you. I meant elements ##\omega_i## of sample space ##\varOmega## rather than sampled values of ##X##.
When we try to understand what sample is mathematically we have two options -
1. samples are ##n## i.i.d. random variables ##X_i## defined on same sample space ##\varOmega## evaluated on...
Thank you. Please note that ##\omega_i## are elements of set ##\varOmega## where we have a field of subsets and a probability measure. They are not random variables with values in reals therefore ##\omega_i## don't have PDFs.
My point is that if every ##\omega_i## does not have same probability...
Please could you help me find a rigorous mathematical definition of sampling as it is used in mathematical statistics?
Let ##X:\Omega\rightarrow\mathbb{R}## be a random variable and ##X_{1},...,X_{n}## is a statistical sample. What is its mathematical meaning in probability theory? Are we...
FYI - http://www.essex.ac.uk/maths/people/fremlin/mt.htm - I have not read it myself and don't know how well it is written.
I read "Measure Theory" by Paul Halmos and found it extremely well written.
I am reading Protter's book on stochastic integration and differential equations.
On a couple of occasions the author used some properties of characteristic functions I am not familiar with. I would appreciate if someone could point me in a right direction.
1. p22 T31 - We are working with...