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I'm given this problem:

A device that continuously measures and records seismic activity is placed in a remote

region. The time, T, to failure of this device is exponentially distributed with mean

3 years. Since the device will not be monitored during its first two years of service, the

time to discovery of its failure is X = max(T, 2) .

Determine E[X].

Solution: 2 + 3 Exp[-2/3]

I've even got the procedure

It's:

E[X] = Integral From 0 To 2 [ 2*f(t)dt ] + Integral From 2 To Infinity [ t*f(t)dt ]

Where f(t)=1/3 Exp[-1/3 t]

I Just want to know, why is this?... Why the interval of the first integral is from 0 To 2, and then again that "2" appears in the integral?... I tried to calculate by means of order statistics but result didn't match

Does someone know how to prove this is actually the solution?... (I'm certain this is the correct solution, but I just want a more specific and justified procedure. I'm not familiarized with constant random variables)

Thanks in advance

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# Expectation Of The Maximum When One Of The Random Variables Is Constant

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