I have just started probability and I am rather confused at the notion of probability measure $P$ and how it essentially differs from the distribution of a random variable $X$ under $P$
First, I need an explanation:
In most textbooks, it is stated for a random variable $X$ from a
Probability space $(\Omega, \mathcal{F}, P)$ to an event space
$(\Omega',\mathcal{F}')$ then $P'(A'):=P_{X}(A')=P(X^{-1}A')$ is a
probability measure, namely the distribution of a random variable $X$
under $P$. But how is this any different from the original Probability
measure $P: \mathcal{F} \to [0,1], A \mapsto P(A)$, which is simply
mapping subsets of $\Omega$ onto $[0,1]$. Is it a different $P'$
probability measure altogether? I mean surely it depends on the map of
our previous probability measure $P$? What is the intuition behind
this rather opaque version of defining the probability with respect to
a random variable?
My example for independence:
Let there be $3$ $6-$sided dice with each dice taking on either colour blue, green or red.
Let event $A:=$ the dice selected is red.
and event $B:=$ the dice selected and then thrown shows $2$.
Let's then take $2$ probability measures $P, Q$. We set $P$ equal to uniform distribution.
It follows that $P(A \cap B) = \frac{1}{6 \times 3}=\frac{1}{18}$
and $P(A) \times P(B)=\frac{1}{3} * \frac{1}{6} = \frac{1}{18}$, so $A, B$ are naturally independent.
Now I am attempting to find a probability measure $Q$ such that:
$Q(A \cap B) \neq Q(A) \times Q(B)$
My idea: perhaps $Q(C)=\frac{2|\Omega|-|C|}{|\Omega|}$. But how can I be sure that it is $\sigma-$additive, it is clearly "normed" as $Q(\Omega)=1$.
$Q(A \cap B)= \frac{2 \times 18-1}{18}>1$, which is not true for a probaility measure, right? Is my example wrong? Can you provide me with more intuitive examples?
Best Answer
The simplest example might be when $\Omega=\{1,2,3,4\}$, $\mathcal F=2^\Omega$, $A=\{1,2\}$, $B=\{1,3\}$.
Then, the events $A$ and $B$ are independent under the uniform measure $P$ but not under the measure $Q$ defined by $Q(\{1\})=\frac12$ and $Q(\{2\})=Q(\{3\})=Q(\{4\})=\frac16$.
Regarding the explanation you demand, simply note that, in the context you describe, $P$ is a measure on $\mathcal F$ while $P'=P_X$ is a measure on $\mathcal F'$. The most common case might be when $(\Omega',\mathcal F')=(\mathbb R,\mathcal B(\mathbb R))$ and $X:\Omega\to\mathbb R$. Then the distribution of $X$ is a measure on the sigma-algebra $\mathcal B(\mathbb R)$ on the target set $\mathbb R$, and certainly not a measure on the sigma-algebra $\mathcal F$ on the source set $\Omega$.
To sum up, in most cases, $P$ and $P'$ are probability measures on different spaces, hence one definitely cannot use one for the other.