A given engineering component lasts t years before failing. I'm having trouble calculating the mean time before failure, starting with a known CDF.
Given a cumulative density function (CDF) as follows, where x>0:
$F(x)=1-\frac{1}{(1+x)^2}$
Deriving a probability density function (PDF) from this, I get:
$f(x)=\frac{2}{(1+x)^3}$
Probability of the component still working after 5 years, I've calculated to be:
$\int_{5}^∞ \frac{2}{(1+x)^3} dx=2.77\%$
Calculting mean time to failure (MTTF) is done by:
$E(Y)=\int_{0}^∞ x f(x) dx$
To which I get t=1 when plugging in my PDF:
$\int_{0}^∞ x \frac{2}{(1+x)^3} dx=1$
What am I doing wrong? I'm worried that I'm conflating the wrong variables as 1 seems wrong given that after 1 year, only 25% of the time will a component be working.
Your calculation is correct.
Computing the expectation gives only a partial intuition of what happens. If you consider a random variable $X$ which is equal to 0 with probability $0.99$ and is equal to $10^4$ with probability $0.01$ has expectation equal to $10^2 = 100$, even if in the 99% of cases it will be equal to 0, on average.
The expectation is a "weighted average" of all the possible outcomes of your random variables, weighted with their probabilities. If there are large deviations, the expectation will be high.
This large deviation is the possibility to take very large values with respect to their probabilities.