# What is an induced probability function?

My textbook defined the probability function of a random variable as:

the function $$P_X$$ is an induced probability function on $$X(\Omega)$$, defined in terms of the original function P.

In other words, the function $$P_X$$ induces a surjection by restricting its co-domain to the image of its domain $$X(\Omega)$$?

Am I correct in stating this?

edit: Textbook is Statistical Inference by Casella and Berger page 29

• For completeness sake: Could you add a reference to the textbook? Feb 27 '20 at 14:15
• I've never seen the wording "probability function". Given the ambient probability space $(\Omega,\mathfrak{F},\mathbb{P})$, the probability induced by $X$ is the law of $X$ (under $\mathbb{P}$), or the distribution of $X$. This is also called the image probability of $\mathbb{P}$ by $X$ (@whuber says push-forward). Feb 29 '20 at 10:29
• @StéphaneLaurent The textbook I'm using doesn't touch upon probability spaces. It is Casella and Berger' Statistical Inference. It defines the probability measure P, which must follow the Kolmogorov Axioms. Feb 29 '20 at 16:01

Not quite. The setting is a probability space $$(\Omega,\mathfrak{F},\mathbb{P})$$ and a measurable function $$X$$ whose domain is $$\Omega$$ and whose codomain usually is $$\mathbb{R}$$ with its Borel sigma-algebra $$\mathfrak{B}$$ (but generally could be any measurable space).

$$X$$ induces a probability distribution $$\mathbb{P}_X$$ as the push-forward of $$\mathbb{P}$$ via $$X$$, sometimes written $$X_{*}\mathbb{P},$$ defined as

$$\mathbb{P}_X(E) = (X_{*})\mathbb{P}(E) = \mathbb{P}(X^{-1}(E)) = \mathbb{P}\left(\{\omega\in\Omega\mid X(\omega)\in E\}\right)$$

for any event $$E\in\mathfrak{B}.$$

Let's do a simple example. Let $$\Omega$$ be the set of the three possible ways a flipped coin may land: heads, tails, or on its edge. Let its sigma-algebra $$\mathfrak{F}$$ consist of all subsets of $$\Omega.$$ Let the probability distribution $$\mathbb{P}$$ assign the value $$p$$ to $$\{\text{Heads}\},$$ $$1-p$$ to $$\{\text{Tails}\},$$ and $$0$$ to $$\{\text{Side}\}.$$ This determines $$\mathbb P$$ on every subset of $$\Omega$$ according to the laws of probability.

The function $$X:\Omega\to\mathbb{R}$$ that equals $$1$$ for $$\omega=\text{Heads}$$ and otherwise equals $$0$$ is the indicator of $$\text{Heads}.$$ $$X$$ is obviously measurable (because every subset of $$\Omega$$ is measurable). To figure out what $$\mathbb{P}_X$$ is, let $$E\subset\mathfrak{B}$$ be a Borel-measurable set. $$X_{*}\mathbb{P}(E)$$ is the sum of up to three values: $$p$$ if $$X(\text{Heads})\in E,$$ plus $$1-p$$ if $$X(\text{Tails})\in E,$$ plus $$0$$ if $$X(\text{Side})\in E$$.

One convenient way to express $$\mathbb{P}_X$$ uses the "one-point" measures $$\delta_a$$ defined on the Borel sets of $$\mathbb{R}.$$ These assign the value $$1$$ to an event $$E$$ when $$a\in E$$ and otherwise assign the value $$0.$$ It's easy to check that they are indeed measures.

The random variable $$X$$ thereby pushes $$\mathbb P$$ into the induced measure (or "induced probability function") $$\mathbb{P}_X = (1-p)\delta_0 + p\delta_1.$$

Another description of the induced measure considers only events of the form $$E(x)=(-\infty, x]$$ for $$x\in \mathbb{R},$$ because these determine the entire Borel sigma algebra of $$\mathbb R.$$ The formula $$F_X: x\to \mathbb{P}_X(E(x)) = \mathbb{P}(X\le x) = \mathbb{P}\left(\{\omega\in\Omega\mid X(\omega)\le x\}\right)$$

defines a function on $$\mathbb R,$$ the cumulative distribution function of $$X.$$ It equals $$0$$ for $$x\lt 0,$$ jumps up to a constant value of $$1-p$$ for $$0\le x \lt 1,$$ and then jumps (by an amount $$p$$) up to $$1$$ for $$x\ge 1.$$

In this example of a Bernoulli$$(p)$$ random variable, please notice that

• $$X$$ is neither an injection nor a surjection from $$\Omega$$ to $$\mathbb R.$$ Its image is merely the set $$\{0,1\}.$$

• $$F_X$$ is neither an injection nor a surjection from $$\Omega$$ to the set of possible probabilities $$[0,1].$$ Its image is the set $$\{0,1-p,1\}.$$