please answer the following question with all the step. Thank you

4. Vehicles arrive at Joe's Bar and Grill according to a Poisson process, NV(t), with intensity o, The number of individuals in the j'th arriving vehicle is a random variable, Y,, such that P(Yj = k) = PK, k = 1,2, ...,6. The Yj's are i.i.d. All individuals in the j'th arriving vehicle enter and leave together and stay in the restaurant a random amount of time, Z,, where the Zi's are i.i.d. exp() = 1/30), thus E(Z;) = 30. Time is measured in minutes. Assume that {N(t) }co, {Yjf, and {Z}}fe, are independent. Define X(t) = the number of individuals that have arrived in (0, and Y (t) = the number of individuals that are in Joe's B&G at time t. Assume that Y (0) = 0. Evaluate, in terms of 6, {px}=1 and t, (a) px(t) ( u), (b) P(X (t) = 0), ( c ) by(t) ( u ) (d) E(Y (t) ) and lim E(Y (t) ). t-+ 0o