Question

1) Let X be the life (in hours) of a certain electronic device. The pdf of X is f(x)- e-100, for x ? 0 and 0 otherwise. What is the expected life of this device? 100 2) The density function of coded measurements of the pitch diameter of threads of a fitting is 4 0 elsewhere. Find the expected value of X

Thank you!

Answer 1

1)

epxected value E(X)=$\int_{0}^{\infty }$x f(X) dx =$\int_{0}^{\infty }$(x/100)e^-x/100 dx

=(-(x/100)*100*e^-x/100-(1/100)*100²*e^-x/100))|^$\infty$₀ =100

2)

expected life =E(X)=$\int_{0}^{1}$ x f(x) dx =$\int_{0}^{1}$(4/$\pi$)*(x/(1+x²)) dx

let (1+x² ) =t

differentiating above with respect to x:

2x =dt/dx

xdx =dt/2

for x=0 ; t=1

and for x=1 ; t-2

theefore E(X)=$\int_{0}^{1}$(4/$\pi$)*(x/(1+x²)) dx =$\int_{0}^{1}$(4/$\pi$)*(1/2)*(1/t)* dt =(2/$\pi$)*(ln(t)|²₁ =(2/$\pi$)*(ln(2)-ln(1))

E(X)=2ln(2)/$\pi$