Can anybody explain the basic difference between Random Variable and Scalar random variable. I know it's a silly doubt, But I couldn't find the answer on internet.
[Math] Difference between Random Variable and Scalar random variable
random variablesstatistics
Best Answer
Suppose we have a probability space $(\Omega, \mathcal{F}, P)$.
A random element is any (measurable) function $X:\Omega\rightarrow S$ where $S$ is some measurable space. Depending on which space $S$ we are working with, we get random variables, random vectors, random processes...
For example, if $S=\mathbb{R}^n$, then $X:\Omega\rightarrow\mathbb{R}^n$ is an n-dimensional random vector.
If $S=\mathbb{R}$, then $X:\Omega\rightarrow\mathbb{R}$ is (real) random variable. If $S=\mathbb{C}$ we would have a complex random variable. In this terminology, we assume that a random variable is a random element which takes values from some scalar field (eg. $\mathbb{R}$). In that sense, a random variable is always scalar, thus same as a scalar random variable.
However, some sources define a random variable to refer to what I defined here as a random element, so in that case a scalar random variable would be what I call here just a random variable - a function on the probability space that takes only scalar values.