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I understand what the law of a random variable is by definition (below), but I don't understand the intuition.

Definition: The law of a R.V. $X$, denoted $\mathcal{P}_X$ is the probability measure on $(\mathbb{R}, \mathcal{B})$ such that $\mathcal{P}_X(B) = \{ \omega: X(\omega) \in B \}$ for any Borel set in $\mathcal{B}$.

Without too much jargon, can someone please tell me the motivation and intuition behind this concept and definition?

FountainTree
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  • $B$ represents an event, and $\mathcal B$ is the set of events that you might reasonably want to consider including $\Omega$ (every possibility) and $\emptyset$ (no possibility). $\mathcal{P}_X(B)$ measures the probability of the event $B$ and $\mathcal{P}_X$ has the three probability properties you want (non-negative, $\mathcal{P}_X(\Omega)=1$ and countable unions of mutually disjoint events have probability equal to the sum of the probabilities of the parts). – Henry Oct 18 '22 at 09:17

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Suppose the probability space $(\Omega,\mathcal{F},\mathbb{P})$ models some random phenomenon you are interested in studying. In many circumstances, the sample space $\Omega$ is not easily dealt with because its elements are not numbers. So the idea is to construct a function $X:(\Omega,\mathcal{F},\mathbb{P})\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$ which transfers the problem of computing the probability associated to events $A$ from $\mathcal{F}$ to the problem of computing the probability of events $B$ from $\mathcal{B}(\mathbb{R})$.

In order to make such transference, we define the induced probability measure $\mathbb{P}_{X}$ as the set function expressed by $\mathbb{P}_{X}:\mathcal{B}(\mathbb{R})\to[0,1]$ where $\mathbb{P}_{X} = \mathbb{P}\circ X^{-1}$. For the purpose of well-definiteness, we have to require that $X$ is a measurable function, otherwise we could arrive at sets in $\Omega$ which are not possible to assign probability. Even more: $X^{-1}(\mathcal{B}(\mathbb{R}))$ is the smallest $\sigma$-algebra that turns $X$ a measurable function. Once you have the probability space $(\mathbb{R},\mathcal{B}(\mathbb{R}),\mathbb{P}_{X})$, you can "forget" in some sense the original space and study the random phenomenon through the induced space.

Átila Correia
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    I think if the definition the OP referenced has too much jargon, this definitely has too much jargon. – Andrew Oct 18 '22 at 04:43
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A random variable is a ("nice") function $X:\Omega \to \mathbb R$, where $(\Omega,\mathbb \mu)$ is a probability space. Given any ("nice") subset $B\subset \mathbb R$, we can look at $\mu(X^{-1}(B))$. This induces a probability measure on $\mathbb R$, which we define as $\mathbb P(B) = \mu(X^{-1}(B)) = \mu(X\in B)$. Unfortunately, it's somewhat difficult to motivate without some measure theory. But basically the idea is this. We want to calculate $\mathbb E X$ which is defined as $\int_{\Omega}X\,d\mu$. However, how exactly do you calculate such a thing? We want to be able to do calculus, and there is a theorem called change of variables which says that $\mathbb E X = \int_{\mathbb R} x\, d\mathbb P$. In particular, if $X$ admits a density, then this is $\mathbb E X = \int_{\mathbb R}xf_X(x) \,dx$, which you are probably familiar with.

Andrew
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  • Several questions: (1) a minor note about terminology: do you mean that $(\Omega, \mu)$ is a measurable space rather than a probability space which is a triple? (2) so is the gist of it that probability laws let us handle the expectation of RVs that don't have density? – FountainTree Oct 18 '22 at 21:25
  • @FountainTree I am thinking of $(\Omega,\mu)$ as a "probability space" where perhaps I should really write $(\Omega,\mathcal F,\mu)$, but because I ignored measurability issues in my discussion, I omitted the event space $\mathcal F$. There is a result that any right-continuous non-decreasing function $F$ with $\lim_{x\to\infty}F(x) = 1$ and $\lim_{x\to -\infty}F(x) = 0$ corresponds to a probability measure $\mathbb P$ on the real line, where $\mathbb P(a,b] = F(b)-F(a)$, and in fact this correspondence is one-to-one. – Andrew Oct 18 '22 at 22:10
  • This gives rise to the following: We have $X:\Omega \to \mathbb R$ is a random variable. To "describe" $X$, it suffices to prescribe such a function $F$, from which we obtain $\mathbb P(X \in B) = \int_B dF(x)$. And there is more. When we define $\mathbb E X = \int_{\Omega} X, d\mu$, it is not at all obvious how to calculate such a thing. The change of variables theorem allows us to write $\int X,d\mu = \int_{\mathbb R} x,dF(x)$. In particular, if $F = \int_{-\infty}^x f(x),dx$ then we have $\mathbb E X = \int_{\mathbb R} xf(x),dx$. – Andrew Oct 18 '22 at 22:15
  • Sorry, it should read $F(x) = \int_{-\infty}^x f(t),dt$ – Andrew Oct 18 '22 at 22:38