What does it mean when we say two events are independent?
What is the difference between two events that are independent and two events that are mutually exclusive?
Why can two events not be simultaneously independent and mutually exclusive?
What does it mean when we say that two random variables are independent?
Why is it not possible for two random variables to be mutually exclusive?
What does it mean when we state that two random variables are identical?
The event $X=x$ is said to be independent of the event $Y=y$ if we can say the following about the conditional probability that $X=x$ given $Y=y$:
$$ P(X=x|Y=y) = P(X=x) $$
It is important to understand the difference between what it means when we say two events are independent and when two events are mutually exclusive. If you do not understand the distinction between independent events and mutually exclusive events you cannot possibly understand anything about probability theory. The distinction is explained pictorially in the video below**:**
https://www.youtube.com/watch?v=1P8stG7g3SU
The random variable $X$ is said to be independent of $Y$ if the value of $X$ does not depend on the value of $Y$. For $X$ to be independent of $Y$ we must therefore have:
$$ P(X=x|Y=y) = P(X=x) \qquad \forall \quad x, y $$
Saying that two random variables are mutually exclusive makes no sense.
Lastly, note that identical random variables are not necessarily independent
<aside> 📌 SUMMARY: When the random variable $X$ is independent of $Y$ the value of $X$ does not dependent on $Y.$ This meaning of this qualitative statement can be made precise by using conditional probability.
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