Do IID random variables take same number of values?

Question

Consider the following line from the definition of IID random variables

If $X_1, \cdots ,X_n$ are independent and each has the same marginal distribution $\cdots \cdots $

The expansion of IID is

independent and identically distributed

I did not understand the bolded part.

What does it mean by identically distributed or same marginal distribution? How to validate that two random variables are identically distributed (like multiplication rule for independent random variables $P(X_1,X_2)= P(X_1)P(X_2)$)?

My current understanding:

$X_1, X_2$ takes same number of values. Let us say $\{a,b,c\}$ for $X_1$ and $\{c,d,f\}$ for $X_2$. Then for some order

$P(X_1 = a) = P(X_2 = c)$, $P(X_1 = b) = P(X_2 = d)$, $P(X_1 = e) = P(X_2 = f)$

I am pretty sure that my interpretation may go too wrong.

Answer 1

Identically distributed means $P(X_i \leq x)$ does not vary with $i$. In the case of random variables taking only finitely many values it is true that they assume not only the same number of values but also the same exact values with same probabilities.

If $X$ takes the values $0$ and $1$ with probabilites $\frac 1 2$ each and $Y$ takes the values $0$ and $1$ with probabilities $\frac 1 3$ and $\frac 2 3$ they ther are not identicaly distributed.

Answer 2

Every random variable $X$ induces a (unique) distribution which is a probability measure on measurable space $(\mathbb R,\mathcal B(\mathbb R))$ where $\mathcal B(\mathbb R)$ stands for the collection of Borel subsets of $\mathbb R$.

If $X$ is defined on probability space $(\Omega,\mathcal A,P)$ then the distribution is prescribed by:$$B\mapsto P(X\in B)$$This for $B\in\mathcal B(\mathbb R)$.

Two random variables $X$ and $Y$ are identically distributed iff they induce the same distribution.

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If $X_1,\dots,X_n$ are said to be independent then this implies that they are defined on the same probability space so that "identically distributed" boils down to "$P(X_i\in B)$ does not depend on $i$ for every $B\in\mathbb B(\mathbb R)$".

Looking at $X:=(X_1,\dots,X_n)$ as a random vector this comes to the same as: "the marginal distributions of $X$ are all the same".