I need somebody to help me to understand the following concepts:
Assuming $X$, $Y$ are random variables (r.v.'s). What does the following represents:
- $P(X+Y|Y)$, what is this?
- $P(X+Y|Y=y)$, my understanding it is a r.v..
- $P(X+Y=s|Y)$,my understanding it is a r.v..
- $P(X+Y=s|Y=y)$, my understanding it is a number.
How do they relate to each other?
Also, in case of Y a continuous r.v., consider $P(X+Y|Y=y)$, but $P(Y=y)$ is always zero. How can this conditioning be thought of?
Thanks a lot.
Edit
I haven't seen people giving answers to my question: in case of $Y$ a continuous r.v., consider $P(X+Y|Y=y)$, but $P(Y=y)$ is always zero. How can this conditioning be thought of? Just to give another example (in addition to the one I gave in comments about the uniform distribution), consider standard Brownian motion. $Pr(B_t\ge a|B_s=b)$ is clearly sensible and different from $Pr(B_t\ge a)$ for $t\ge s$. But here $Pr(B_s=b)$ is zero.
Also I read in books on "Markov Chains", for example, the notation of Markov property is stated as: $Pr(C_t|C_{t-1}, ..., C_1)=Pr(C_t|C_{t-1})$ So there is the notation $P(X)$ or for that matter, $P(X|Y)$. Is this notation short for $P(X=\text{any value}|Y=\text{any value})$?