Problem 35 Solution

Warning: This is just a quick explanation which assumes some knowledge of the exponential distribution and integral calculus.

The density function of the shorter lived life bulbs is f(x)=1/100 * e^-x/100.

The density function of the shorter lived life bulbs is f(x)=1/200 * e^-x/200. The survival is 1 minus the integral of the density function. S(x)=e^-x/100, and S(x)=e^-x/200.

The survival function for the entire group is (e^-x/100)⁵ * (e^-x/200)⁵.

The density function for the group is the derivitive of 1 minus the survival function = e^-3x/40.

The mean for the group is the integral from 0 to infinity of x times the density function for the group = 40/3 hours.

Michael Shackleford, A.S.A.