Example of a Probability Space for a continuous random variable to help me understand the concept

Question

Probability space is always described very abstract in literature so I seek a specific example to ease my understanding of the concept.

Formal definition: Let $X$ be a real-valued random variable defined on the probability space: $(\Omega,\mathbb{F},P)$

Construction of the example: Now let us assume that $X$ is standard normal so CDF and PDF are known. In this example I want to know that the variables are:

• What is $\Omega$ then?
• What is $\mathbb{F}$?
• What is $P$?

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Wikipedia have a great example of a discrete case with dice.I want a similar explanation for a continuous variable :)

Answer 1

Why not start with something simpler than the normal distribution. For example, suppose the random variable $X$ has a uniform distribution on $[0,1]$.

Then the sample space $\Omega$ is the set of points in the interval $[0,1]$.

The 'events' of $\mathbb{F}$ are things like $0.5\le X \le 0.8$.

$P$ assigns probabilities to events, for example $P(0.5\le X \le 0.8)=0.3$.

Does this help?

The normal distribution

In effect this is a very similar example to the one above.

We must replace $[0,1]$ by $(-\infty, \infty )$ and it is useful to consider the function $P$ in terms of areas under a curve.

Answer 2

Let $X$ be a real-valued random variable defined on the probability space: $(\Omega,\mathbb{F},P)$

This means that $X$ is considered a measurable map $\Omega\rightarrow E$ into an observation space $E$ endowed with a corresponding $\sigma$-Algebra $\mathbb{E}$.

(For real-valued random variables a usual choice for the observation space is $E=\mathbb{R}$ and $\mathbb{E}$ being the Borel-$\sigma$-Algebra on $\mathbb{R}$).

This setting induces a probability measure $P^X$ on the measurable observation space $(E,\mathbb{E})$ by $$ P^X(B)=P(X\in B)=P(\{\omega\in\Omega:X(\omega)\in B\}) $$

for any $B\in\mathbb{E}$.

Saying that $X$ is a standard normal random variable now means that $P^X((-\infty,x])$ is the standard normal CDF.

To get such a standard normal random variable one might choose

• $\Omega=[0,1]$,
• $\mathbb{F}$ being the corresponding Borel-$\sigma$-Algebra and
• $P$ as the uniform measure on $\Omega$.

This assumes that your source of randomness is a uniform variable on $[0,1]$.

One can now define $X$ as the measurable map $\omega\mapsto\Phi^{-1}(\omega)$ with the standard normal CDF $\Phi$. This ensures that $$ P^X((-\infty,x])\\ =P(\{\omega\in\Omega:X(\omega)\leq x\})\\ =P(\{\omega\in\Omega:\Phi^{-1}(\omega)\leq x\})\\ =P(\{\omega\in\Omega:\omega\leq \Phi(x)\})\\ =\Phi(x) $$ is in fact the CDF of the standard normal distribution.

The above choice of $(\Omega,\mathbb{F},P)$ is quite arbitrary: In many cases the particular source of randomness (that is: the probability space $(\Omega,\mathbb{F},P)$) is not of any interest and might even be ignored as long as the observation space $E$ and the induced probability measure $P^X$ have the intended properties.