MTTF is the area under the reliability curve

The reliability curve, also known as the survival graph eventually approaches 0 as time goes to infinity. How do we show that the area below the reliability curve is equal to the mean time to failure (MTTF) or average life of the item?

We would like to show that the expected or average life of an item is:

$E(t)=\int_{0}^{\infty}R(t)dt$

Integrate by parts:

$\int udv = uv -\int vdu$
$u=R(t)$
$dv=dt$
$\frac{du}{dv}=\frac{dR(t)}{dt}$

But $R(t)=1-F(t)$ and $\frac{dF(t)}{dt}=f(t)$

Therefore $\frac{du}{dt}=\frac{-dF(t)}{dt}=-f(t)$

Then $du=-f(t)dt$

Substituting into $\int udv = uv -\int vdu$ :

$\int_{0}^{\infty}R(t)dt=R(t)t |_{0}^{\infty}-\int_{0}^{\infty}t(-f(t))dt$

But the first term evaluates to 0 because the reliability at infinity is 0. Therefore:

$\int_{0}^{\infty}R(t)dt=\int_{0}^{\infty}t(f(t))dt=E(t)=MTTF$

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MTTF is the area under the reliability curve

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