Suppose that two electronic components in the guidance system for a missile operate independently and that each has a length of life governed by the exponential distribution with mean 3 (with measurements in hundreds of hours). Find the probability density function for the average length of life of the two components.

attached below

Step-by-step explanation:

Assuming that Y1 and Y2 are independent exponential random variable with mean 3

where : U = Y1 + Y2 applying the property of exponential distribution sum of exponentials will follow a gamma distribution

i.e. ∝ = 2 , and β = 1

note : U = 2X

Hence probability distribution function X =

Fx (x) = p( X ≤ x )

= p( U ≤ 2x )

= Fu (2x)

probability density function  X

f (x) =  = 2 Fu (2x)

attached below is the remaining part of the solution