Say you have a random variable, X.
Suppose g(x) = x+3 for all x. In this case, g(X) is a random variable that takes a value g(x) when X=x.
I am a bit confused with this. g(X) is a function of X, and g(x) is a function of x. What is the difference between these statements? Essentially, they say the same thing?
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Heuristically they are virtually the same, but more precisely, a random variable is a (measurable) function from a sample space of "outcomes" to a measurable space e.g., the real numbers. For example, $X$ could be the RV that which is the height of a randomly chosen person in a population. Since you could think of this as a function f(person) = height, it's easy to blur the distinction. Then if $g(x) = x + 3$ (function) you can think of $g(X)$ as "pick a random person and take their height so now $X=x$ (RV $X$ takes on the value $x$) and then add $3$". This would be the meaning of $g(X)$ in this context. Not my field so there are probably fine points omitted but generally I think of it this way. Note that if $X$ is real-valued, that implies a probability distribution which must take values in $[0,1]$ and leads to statements like "Let $X$ be a normal random variable".
When you think about X, you often think about an object that symbolizes complete information about some random experiment, such as rolling a dice. When you think about x, you often think about a concrete outcome of this experiment.
Now, sometimes you want to take an existing random experiment and change it in some way. Let us denote the "change" we want to make as (g). You want to make this change in a well-defined manner, so it's better to represent it as a function.
The new experiment you have here is g(X). Now, if you want to consider the outcome x in the new experiment, you denote it as g(x).