I'm reading a text book about probability and this syntax $P(X=n)$ keeps resurfacing. I've googled around, but I keep getting results on binomial distributions.
What does $P(X=n)$ mean? Does it mean that a certain value $X$ has a variance of $n$? Does it perchance have something to do with a concept called a "random variable"?
Best Answer
You can think of a random variable $X$ as a function that maps events to real numbers, i.e., $X : \Omega \rightarrow \mathbb{R}$. The expression $P(X=n)$ denotes the probability that a random variable $X$ has value $n$. Formally, it is a short form for the probability of the event $\{\omega : X(\omega)=n\}$, i.e., $$P(X=n) = P(\{\omega : X(\omega)=n\}).$$ This is also explained here.
For example, if your events are the outcome of throwing a die, you would write the probability of getting a $3$ as $P(X=3)$.