Suppose we have some financial data, e.g., stock return time series. The theoretical distribution is unknown, while we can construct the empirical distribution through historical data. The problem is to compute the probability that the random process will exceed (i) some $x$ value, and (ii) some $x$ value conditional that current return is equal to $y$ ($y<x$).
Intuitive, the first and second parts of the problem should yield different results. However, mathematically I get the same. Here is my reasoning.
(i): we can compute $P(X>x)$ easily from the empirical CDF.
(ii) mathematically the problem is compute $P(X>x|X=y)$. According to conditional probability, $P(X>x|X=y)=\frac{P(X>x \cap X=y)}{P(X=y)}$.As we know that current return $y$ is less than the threshold value $x$ (i.e., $y<x$), the events $X>x$ and $X=y$ are independent (cause these events cannot take place simultaneously). Therefore, $P(X>x|X=y)=P(X>x)P(X=y)$. As a result, $P(X>x|X=y)=\frac{P(X>x)P(X=y)}{P(X=y)}=P(X>x)$.
Question(s): Do you think my reasoning is correct? Regarding (ii), non-intuitive for me result is that probability that return will exceed some $x$ is the same, regardless the condition. I assume that the conditional probability should depend on the current return.
Your computations are correct but the conclusions ar wrong.If two events cannot occur simultaneously they are not independent but mutually exclusive, i.e. they are the null event. That event has zero probability of occuring.
You can think it like this: "I'm currently at 5. What is the probability of being at 7?" Since I'm at 5 I cannot be at 7 as well, hence the probability is $0$