We are given ℎ(푡) as the following pulse function: ℎ(푡)=0.5 at 0≤푡≤2 and 0 otherwise and define the discrete-valued, continuous-time random process 푍(푡)=ℎ(푡+푇) where 푇 is an exponentially distributed random variable with parameter 휆. (a) Sketch a generic sample realization of 푍(푡) as a function of an arbitrary value of 푇. (b) Find the pmf of 푍(푡) (HINT: Since 푍(푡) is discrete-valued, what values can it take?) (c) Find 푚푍(푡) and 퐶푍푍(푡,푡+푑) for 푑2 and 0

Question

We are given ℎ(푡) as the following pulse function: ℎ(푡)=0.5 at 0≤푡≤2 and 0 otherwise and define the discrete-valued, continuous-time random process 푍(푡)=ℎ(푡+푇) where 푇 is an exponentially distributed random variable with parameter 휆. (a) Sketch a generic sample realization of 푍(푡) as a function of an arbitrary value of 푇. (b) Find the pmf of 푍(푡) (HINT: Since 푍(푡) is discrete-valued, what values can it take?) (c) Find 푚푍(푡) and 퐶푍푍(푡,푡+푑) for 푑>2 and 0<푑<2.

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