Intuitive example for Identical random variables

Question

I intuitively understood the idea of independence between two random variables. But hard in getting sense of identical random variables on the same sample space. I saw many examples for identical events over internet but not on identical random variables.

I understood the concept of identical random variables formally and I can provide an example for identical random variables as follows

Let $\Omega = \{a,b\}$ be the sample space and $X_1, X_2$ are two random variables over $\Omega$ defined as follows

$X_1(a) = 0.3, X_1(b) = 0.7$

$X_2(a) = 0.7, X_2(b) = 0.3$

Now, I can say that $X_1, X_2$ are identical because of same range and CDF.

But I am not getting any real life intuitive example such that two identical random variables.

Answer 1

Identically distributed random variables are just random variables that have the same pdf. Here are some examples:

Joe flips a coin 100 times and then writes down the number of heads that come up. Let's call this variable $X_{\textrm{Joe_Heads}}.$ Now Max does the same thing and writes down the number of heads he gets. Let's call this random variable $X_{\textrm{Max_Heads}}.$ Now $X_{\textrm{Joe_Heads}}$ and $X_{\textrm{Max_Heads}}$ are clearly not the same variable. But $X_{\textrm{Joe_Heads}}$ and $X_{\textrm{Max_Heads}}$ are identically distributed because it is equally likely that Joe will flip $n$ heads as it is that Max will flip $n$ heads. These two variables are also independent.

Now as another example suppose that after Joe flips the flips the coin 100 times he writes down the number of tails flipped as well as the number of heads flipped. The number of tails that he flips is now the random variable $X_{\textrm{Joe_Tails}}.$ Once again $X_{\textrm{Joe_Heads}}$ and $X_{\textrm{Joe_Tails}}$ are not the same random variable, but they are identically distributed because the probability that Joe will flip, say, 55 heads is identical to the probability that he will flip 55 tails. And since the number of heads Joe flips determines the number of tails he flips and vice-versa, these random variables are not independent, in contrast to the first example.