Are X and Y independent? Choose your answer below and state the reason on the scratch paper.

Find the marginal distribution of X: P(X=0) = P(X=1) = P(X=2) =  

Compute the average number of customers (waiting or being served) in S1. Show your work on the scratch paper. Compute two digits after the decimal point.

A service station consists of two back-to-back stations S1 and S2. Customers follow this workflow: Customers first arrive at S1​, are served, and then proceed to S2​. After service at S2​, with probability 0.1, a customer is sent back to S1​ for rework (reentrant flow). Otherwise, with probability 0.9, the customer leaves the system. Assume the following conditions: The arrival process follows a Poisson distribution with rate 10 per hour. The service times at S1 and S2​ follow exponential distributions with means 5 minutes and 6 minutes​, respectively. S1 has a single server and S2 has two servers. Both have infinite queue capacity.

Consider a drive-through restaurant where all orders are processed by one order kiosk with virtually unlimited space for the cars to line up. Suppose one car arrive every 3 minutes, while it takes 90 seconds on average to take the order. Show your work on the scratch paper when answering Q20-Q22.

Where did the Briggs v Elliot case take place and why was it important? 

Someone claims that the stationary probabilities for the DTMC is

Describe two key Supreme Court (SCOTUS) decision issued by the Warren Court. 

According to our textbook author, explain two contributing factors to U.S. prosperity during the 1950’s. 

Physical activity is defined as