Joint Random Variables

Question

I am trying to clarify the meaning of a joint pdf.

A collection of random variables $\{X_i\}_{i=1}^{n}$ that are i.i.d. defined on the space $(\Omega, F, P)$, we define their joint probability as:

$P(X_i\in A_i|i\in \{1,2,...,n\})=P(\cap_i^n X_i\in A_i)=\prod_i^nP(X_i\in A_i)$

If they are only identically distributed then each $X_i$ need not be defined in the same space, only require $P_i(X_i\in A_i)=P_j(X_j\in A_j)$. Then $\{X_i\}_{i=1}^{n}$ is defined on $(\Omega_i\times\Omega_j\times...\times\Omega_n, F_i\otimes F_j\otimes...\otimes F_n,P_i\otimes P_j\otimes...\otimes P_n) $ Then the joint pdf is a product measure defined as: $P_i\otimes P_j\otimes...\otimes P_n(A_i\times A_j \times...\times A_n)=\prod_i^nP_i(X_i\in A_i)$

How is the probability calculated for the situation where they are independent but not identically distributed?

Is my understanding correct? I am trying to determine how to calulate the joint probability under different scenarios.

Answer 1

If $X_i$'s are identically distributed but not necessarily independent you cannot calculate the joint distribution in terms of their individual distributions.

If they are independent then $P(X_1 \in A_1,X_2 \in A_2,..., X_n \in A_n)=\prod_{i=1}^{n} P(X_i \in A_i)$.

If they are i.i.d. then $P(X_1 \in A_1,X_2 \in A_2,..., X_n \in A_n)=\prod_{i=1}^{n} P(X_1 \in A_i)$.

Answer 2

Random variables are the same thing as measurable functions. $P$ is a probability measure on $\Omega$. Independent random variables $\{X_i\}_{i=1}^n$ are just $n$ different measurable functions from $\Omega$ to $\mathbb R$ such that the event $$ \{\omega\in \Omega\colon X_i(\omega)\in A_i\ \forall\ i=1,\ldots,n\} $$ has $P$-measure equal to $$ \prod_{i=1}^n P(X_i\in A_i), $$ for all collections of measurable subsets $A_1,\ldots,A_n$ of $\mathbb R$.

Whether or not the $X_i$ have the same distribution has no bearing on the definition of independence.