Question

A3. 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.
Calculate the probability that the lifetime of the machine part is less than 10 years.
Hint: Show that f(t) is legitimate and find the proportionality constant.

Answer 1

f(t)=c(t+10)² for 0<x<40

1=$\int$⁴⁰₀c(t+10)²dt=c$\int$⁴⁰₀t² dt+c ∫⁴⁰₀20tdt+c ∫⁴⁰₀100dt=c(64000/3+16000+4000)=c*41333.3

c=1/41333.33=2.419*10^-5

P= ∫¹⁰₀c(t+10)²dt=c(1000/3+1000+1000)=c*2333.33=0.056