This is a proof that I developed for myself and I'm asking whether it is true.
Let X be a random variable that take integer values with probability mass function of PX(x). Let Y be a random variable that also take integer values.
Consider the below discrete random variable,
PX(Y)
We can describe this as follows,
Let b be an event such that Y(b) = z. The random variable PX(Y) describes the probability of event a that corresponds to X(a) = z.
Now I tried to solve the probability mass function of PX(Y) for a given value.
Which is PPX(Y)(0.1).
My argument is, PPX(Y)(0.1) tells us to find the probability of such event c.
To find this event c, I thought it as the event which gave PX(Y) = 0.1. That means there is such an event b in Y's sample space and an event a in X's sample such that X(a) = Y(b). And P(a) = 0.1.
Therefore the event that gave the random variable 0.1 value is a. Hence c = a.
Therefore PPX(Y)(0.1) = 0.1.
Am I correct. If you can point out what I've done is correct or wrong I'm very glad.
Best Answer
I went through some further thinking on the problem and found an elegant solution.
Given above random variables $X$ and $Y$. Since ${P_X(Y)}$ is a random variable, it has a PMF. So my interpretation for $P_ X(Y)$ is follows.
Let $b$ be an event s.t $Y(b) = z$, for this event $P_ X(Y)$ takes the value of $P(a)$, where $a$ is an event in $X$'s sample space s.t $X(a) = z$
Now lets consider the PMF of $P_ X(Y)$, which is $P_{P_X(Y)}$.
If we calculate the $P_{P_X(Y)}(0.1)$, my argument is,
$P_{P_X(Y)}(0.1)$ tells us to find the probability of such event $c$.
To find this event $c$, I thought it as the event which gave $P_ X(Y)$ = 0.1. That means there is such an event $a$ in $X$'s sample space and $P(a) = 0.1$. This event $a$ is found for $P_ X(Y(b))$ such that $X(a) = Y(b)$ for an event $b$ in $Y$'s sample space. Therefore the event of interest that gave the random variable a value of 0.1, is $b$. Hence $c = b$.
Therefore $P_{P_X(Y)}(0.1)$= $P(b)$.