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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.

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  • $\begingroup$ Not a stupid question - I mean I don't know what the difference is! Admittedly I have never heard of a difference, other than Random vector and a random variable. $\endgroup$
    – Chinny84
    Jan 18, 2017 at 19:23
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    $\begingroup$ In the general probability context, a random variable is just a function defined on the probability space, which could have a wide variety of possible codomains. A scalar random variable is a scalar-valued function on the probability space (so typically real-valued, but maybe complex-valued). (Technically any random variable must be measurable, but ignore that if you're not familiar with measure theory.) $\endgroup$
    – Ian
    Jan 18, 2017 at 19:45

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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.

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