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Suppose that $a = 5$. In this case, $a$ is not random since it is always equal to $5$. Can $a$ instead be interpreted as a random variable $X$ such that \begin{equation} p(X=x) = \begin{cases} 1 &\text{if} \quad x = 5 \\ 0 &\text{otherwise} \end{cases} \end{equation} If so, would $a$ be interpreted as a discrete random variable, as indicated by the probability mass function above, or as a continuous random variable?

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    $\begingroup$ Looks like Dirac's delta distribution probabilitycourse.com/chapter4/4_3_2_delta_function.php $\endgroup$
    – Many
    Commented Jun 10, 2021 at 12:26
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    $\begingroup$ Sometimes this is called a "constant random variable." stats.stackexchange.com/search?q=%22constant+random+variable%22 $\endgroup$
    – Sycorax
    Commented Jun 10, 2021 at 15:31
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    $\begingroup$ Label all faces of die "5:" because that does not change the random nature of the outcome of rolling it, perforce it defines a random variable. That variable just happens to have a constant value. Yes, we could declare that such variables are special and agree to not call them "random"--but then the statements of many theorems about random variables would have to be cluttered with exceptions and special cases. $\endgroup$
    – whuber
    Commented Jun 10, 2021 at 20:09

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Given a probability space $(\Omega, \mathscr F, P)$, a random variable is a Borel function $X : \Omega\to\mathbb R$. That's all there is to it.

For $a\in\mathbb R$ let $X_a : \Omega\to\mathbb R$ be defined by $X_a(\omega) = a$ for all $\omega\in\Omega$. Consider a Borel set $B \subseteq \mathbb R$. Then $$ X_a^{-1}(B) = \{\omega\in \Omega : X_a(\omega) \in B\} = \begin{cases}\Omega & a \in B \\ \emptyset & a \notin B\end{cases}. $$ No matter how coarse the $\sigma$-algebra $\mathscr F$ is, it will always be the case that $\{\emptyset, \Omega\}\subseteq \mathscr F$ so this means that $X_a$ is Borel.

The distribution for $X_a$ is $P\circ X_a^{-1}$ and $$ (P\circ X_a^{-1})(B) = \mathbf 1_{a\in B} $$ i.e. any set containing $a$ has a measure of $1$ and any set not containing $a$ has a measure of zero. Effectively this is a point mass of $1$ on $a$, and the corresponding pmf is just $P(X=x) = \mathbf 1_{x = a}$.

This distribution is not absolutely continuous w.r.t. the Lebesgue measure so it does not have a pdf in the usual sense. The Dirac delta is a way to treat it as if it does, but really integrating with respect to a Dirac delta is just using a different measure.

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