Answer to Question #115190 in Statistics and Probability for desmond

Question #115190

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.

Expert's answer

"\\text {Let c is a constant, then}\\\\\nf(t)=c[(t+10)-2]=c(t+8), 0<t<40\\\\\n\\text{Its known that if f(t) is a density function,}\\\\\nThen,\\; \\int _0^{40}f(t)dt=1\\, so,\\\\\nc\\int _0^{40}t+8dt=1\\\\\nc\\left[\\frac{t^2}{2}+8t\\right]^{40}_0=1\\\\\nc[\\frac{40^2}{2}+8(40)]=1\\\\\nc(1120)=1\\\\\n\\therefore c=\\frac{1}{1120}\\\\\nP(t<10)=\\frac{1}{1120}\\int _0^{10}t+8dt\\\\\n=\\frac{1}{1120}\\left[\\frac{t^2}{2}+8t\\right]^{40}_0\\\\\n=\\frac{1}{1120}[\\frac{10^2}{2}+8(10)]\\\\\n=\\frac{1}{1120}\\times 130\\\\\n=\\frac{13}{112}"