Question

The lifetime of a machine is continuous on the interval (0, 40) with probability density
function f, where f(t) is proportional to (t+ 10))2
, and t is the lifetime in years. Find
the proportionality constant that makes the f(t) a probability function.

Answer 1

$\int_0^{40} k(t+10)^{-2}dt=1$.

So, $\frac{-k}{t+10}|^{t=40}_{t=0}=\frac{k}{10}-\frac{k}{50}=\frac{2k}{25}=1.$

Thus, $k=\frac{25}{2}=12.5.$