A problem as following: (from Prob, statistics, and random processes for electric engineering, p.264)
If I want to find $P(Y\leq y \mid X=+1)$, it can be calculated as following: $P[N+1\leq y]$. It is obvious. (As book did)
My question is why not: $\dfrac{P(Y\leq y,X=+1)}{P(X=+1)}$.
How to make sure both of them equal?
$$\frac{P(Y\leq y,X=+1)}{P(X=+1)}=\frac{P(Y\leq y|X=+1)\cdot P(X=+1)}{P(X=+1)}=P(Y\leq y|X=+1)=P(X+N\leq y|X=+1)=P(1+N\leq y)$$ In other words $P(Y\leq y,X=+1)=P(1+N\leq y)\cdot P(X=+1)$