Let's say I know the form of the lifetime pdf for some object class. If I select an arbitrary object from the class which is still alive and for which I have no ancillary information on its current age, is it true that the current age pdf for that object is always equal to the lifetime pdf? I have convinced myself this is true when considering trivial lifetime pdfs such as the uniform distribution, but I am still not sure about the general case.
I have researched several disciplines including survival analysis, but i have not seen this seemingly simple question addressed.
No this is not true. It is an instance of the "inspection paradox". You are more likely to choose an object with a longer lifetime, which makes sense, since the objects with short lifetimes are eliminated from your sample.