What is the difference between multivariate random variables and sample random variables?

Question

Multivariate random variables consists of more than one random variable which may be independent , eg. Height , weight , age can be called three random variables and we can write their joint distribution .These can be represented by X1,X2,....Xn. But when I was reading random sample and estimation topic , I came across random sample variables X1,X2...Xn . Are they same as above or for eg. they denote same random variable height for different samples from ?

Answer 1

They are the same, but usually they denote $n$ iid random variables. I.e., a sample $\{x_i\}_{i=1}^n$ was drawn from a population $X$ that has a distribution $f_X(x;\theta)$. If the draw was random, then the $n$ drawscan be thought as $n$ iid random variables, hence their distribution is just a multiplication of their cumulative distribution functions, i.e., $$ \mathbb{P}(X_1 \le x_1, ..., X_n \le x_n) = \prod_{i=1}^n F_{X_i}(x_i) $$ and the density is $$ \prod_{i=1}^n f_{X_i}(x_i). $$