16 | May | 2023 | Physics Town

Random variable $X$ represents a single-valued result of a random event. Its value is unknown to us, not because of our ignorance, but because of its non-existence. The value exists only after the happening of that random event.

Symbol $x$ represents a particular value of $X$ . It is an existing value that can be substituted to $X$ . We use symbol $x$ instead of a number because we have not yet known what that particular number is.

— Me@2016-04-08 05:24:45 PM

X ~ random variable

It is a variable due to the fact that the “identical” random process can result differently at different times.

x ~ a value of X

Since it is a particular value of X, it is not a variable. However, it may seem to be a variable because it may still be unknown to us.

Symbol $P(X)$ is meaningless because inside, it must be a statement (representing an event). Symbol $X$ is a random variable, not a statement.

Instead, “ $X=x$ ” is a statement. So expression $P(X=x)$ is meaningful, such as

$P(X=x) = {\begin{cases}{\frac {1}{2}},&x=0,\\{\frac {1}{2}},&x=1,\\0,&x\notin \{0,1\} \end{cases}}$

From another point of view, $X$ is a noun phrase, such as “my monthly salary”, not a number. Symbol $x$ is a number, although maybe not known yet. That’s why whatever the formula, it contains no $X$ ‘s, but $x$ ‘s. For example,

$\cdots = {\begin{cases}{\frac {1}{2}},&x=0,\\{\frac {1}{2}},&x=1,\\0,&x\notin \{0,1\} \end{cases}}$