I get the following question from Zastawniak's Probability Through Problems:
Assume that the distribution function of a random variable $X$ on a probability space $(\Omega,{\mathcal A},P)$ is defined as $P^X(A)=\frac{1}{3}\delta_0(A)+\frac{2}{3}P_2(A) $ for any Borel subset $A$ of ${\mathbb R}$, where $\delta_0$ is the Dirac measure and $P_2$ is an absolutely continuous probability measure with density $ f(x)=\frac{1}{2}1_{[1,3]}. $ Show that $P^X$ is a probability measure on $({\mathbb R},{\mathcal B})$.
This can be done by a couple line using the definition of probability measure. Here are my questions:
- $P^X$ seems kind of combine a discrete and a continuous random variable. Can $X$ be written as a linear combination of two random variable $X_1$ and $X_2$ such that $P^{X_1}=\delta_0$, $P^{X_2}=P_2$?
- Can any one come up with a textbook with detailed discussion of such random variables?
- How can I calculate $ \int_{\Omega}XdP? $ One more step I can come up with is $ \int_{\mathbb R}XdP^X $ But I have no idea how to deal with $d(\frac{1}{3}\delta_0(A)+\frac{2}{3}P_2(A))$ theoretically.