Introduction
Random variables are a handy way to deal with the actual probability space, Which may be very complex. There are some usual outcomes and experiments that you study, and practice is quite tricky, and there are a lot of different effects that are of interest to us. There are a lot of other random variables defined on the same probability space that are of interest to us. These random variables could be connected because they all are defined in the same space. So how do we deal with that? What are the tools, and what are the definitions of the distributions? How do we do computation with that? And how do we generally think about such scenarios is the main focus of this blog?
Joint PMF of a single Random variable
So let's begin with a straightforward example. Let's say we have a fair coin, and we toss it three times. Naturally, there can be three random variables.
let if i th toss is heads and
if i th toss is tails, i = 1,2,3.
Together with the three random variables, describe the experiment's outcome entirely.
The event is independent of
and
.
Now, these three random variables are defined in the same space. So when we have repeated trials of any sort, naturally, you will have multiple random variables.
So let's move with a complicated example.
A two-digit number from 00 to 99 is selected at random. Partial information is available about the number as two random variables. Let X be the digit in the tens place. Let Y be the remainder obtained when the number is divided by 4.
We can extract partial information and define the random variables. X can take up the values from 0 to 9, and Y can take up from 0 to 3.
Possible outcomes of X = 0 are {00, 01, 02 , 03,...09} = 10 outcomes.
Similarly, possible outcomes for X=i, are 10.
Possible outcomes of Y = 0 are { 00, 04, 08, 12, 16,...,96} = 25 outcomes.
Similarly, possible outcomes for Y=i, i=0,1,2,3 are 25.
We can define a PMF probability table or probability mass function for each random variable X and Y.
Suppose that event X=1 has occurred; what about the event Y=0? The two events will not be independent; one will determine or affect the occurrence of the other event. So these are different types of random variables dependent on each other.