while solving an integral problem for bivariate distribution, I encounter with some problem when interpreting the results.
Let say that I have a bivariate continuous function $f(t,x)$ where $t$ and $x$ represents time $T$ and biomarker value $X$ respectively. If I solve the following integral (1),
$$(1) \int_{-\infty}^{\infty} f(u,x) du = f_X(x)$$
I will arrive to the definition of the marginal distribution of $f_X(x)$. However, if I solve the following integral (2),
$$(2) \int_{t}^{\infty} f(u,x) du = ??,$$
Do I get probability $P( T>t \cap X=x )$ or $S(t | X=x)$? Here, $S(.)$ is the survival function.
This is because I have read some literature regarding right-censoring and encounter with situation where solving (2) will produce $S(t|X=x)$. But if we recalled the conditional probability, the conditional survival should be as following,
$$ S(t | X=x) = \frac{\int_{t}^{\infty} f(u,x) du}{\int_{-\infty}^{\infty} f(u,x) du}$$ $$ S(t | X=x) = \frac{\int_{t}^{\infty} f(u,x) du}{f_X(x)}.$$
Did I missed out anything? Thank you for the explanation.