6.7.12 Let be the time of the first event in a .0;/ Cox process. Let the 0 and

states represent “OFF” and “ON,” respectively.

(a) Show that the total duration in the .0; ] interval that the system is ON is exponentially distributed with parameter and does not depend on ;;

or the starting state.

(b) Assume that the process begins in the OFF state. Show that the total duration in the .0; ] interval that the system is OFF has the same distribution as

where is exponentially distributed with parameter ;N.t/ is a Poisson process with parameter , and 0;1; : : : are independent and exponentially distributed with parameter .

Transcribed Image Text:

N() M k=0