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

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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))|\infty0 =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

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