I am given a probability space $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mathbb{P})$, where $ \mathbb{P}=0.1 \delta_{-2}+0.7 \delta_{1}+0.2 \delta_{10}$.
I am also given a X$(\omega)=2 \omega^{3} I_{(-\infty, 3]}(\omega)$ and the question is to determine $\mathbb{P}_{X}([0,2])$.
Now, my understanding is that $\mathbb{P}$ is probability measure that maps $\mathcal{B}(\mathbb{R})$ to [0,1].
What I don't understand is how X$(\omega)$ connects $\mathbb{P}_{X}$ and $\mathbb{P}$. I read What is the difference between a probability distribution on events and random variables? but the answer was not very helpful. I read Jacod & Protter and but still cannot figure out how the two are related and honestly feel quite dumb.
P.S. I am also looking for a reference book that is more focused on problem solving than theory to grasp mechanics. So far I tried Çınlar, Jacod & Protter, Shiryaev and Schilling. Schilling is the most gentle but all of them are still more theory focused.
Best Answer
Random variable $X$ induces a new probability measure on $(\mathbb R,\mathcal B(\mathbb R))$.
The measure is denoted as $\mathbb P_X$ and is prescribed by:$$B\mapsto\mathbb P(X\in B)$$ So based on that we find that: $$\mathbb P_X([0,2])=\mathbb P(X\in[0,2])=\mathbb P(\{\omega\in\mathbb R\mid X(\omega)\in[0,2]\})=\mathbb P(\{\omega\in\mathbb R\mid2\omega^2\mathbf1_{(-\infty,3]}(\omega)\in[0,2]\})$$ Now observe that: $$2\omega^2\mathbf1_{(-\infty,3]}(\omega)\in[0,2]\iff\omega\in[-1,1]\cup(3,\infty)$$
So it remains to find:$$\mathbb P([-1,1]\cup(3,\infty))$$Can you take it from here?