Expected value of R.V. or P.D.F. Which one is correct to say I have seen both in many places and I am not really sure.
The expected value of a random variable or of a probability density function?
In similar terms is it true for other measures such as variance?

Technically it is proper to say expected value of a random variable, since that's what you are really looking for. 
We have a variable that spits out different numbers in probability, and we want to know what "number to expect". 
It's as bluntly as I can put it. 
I'd imagine people might say finding "expected value of a pdf/cdf/distribution", which I guess is the "lazy way", and could mislead at times, but people know what others mean implicitly usually. 
When I say misleading, its because there are times when you actually are looking for the expected value of a cdf, say you want to randomize it's parameter:
$$E[F_x(x;Y)]=\int_{-\infty} ^\infty F_X(x;y) f_Y(y) dy$$
This is used when doing mixtures or transformations of r.v.s 
Though for the most part it's not a big deal to say expected value of a distribution so long as the context is okay. 
