I think something with hazard rate / survival function just clicked

These concepts are very confusing, but I think I'm starting to understand the relationships ... is all of the following correct?



F(X) = the probability of X being less than X = x

f(X) = the probability that X will take on the value X = x

s(X) = survival function = probability of taking on a value ABOVE X = x, which implies s(X) = 1 - F(X)

h(X) is more confusing, so I'm not sure here. But I believe it's defined as the probability that X will take on the value of X = x given that X survived until this point = f(X) / s(X)



Now, some useful things that can come from the above which just clicked in my head:



E(X) = integral from 0 to infinity of the survival function (not sure conceptually why this is)

E(X^2) = integral from 0 to infinity of X=x ^ 2 * f(X) (also not sure why this is true)

V(X^2) therefore comes from E(X^2) - E(X)^2



So, with any problems asking for E(X) or Var(X) giving me the S(X) or h(X) functions, I can figure it out just using the formulas above, correct?



Are there any distributions this is more important for than others? IE exponential



Thanks

I think something with hazard rate / survival function just clicked