Consider a collection 4,..., 4, of mutually exclusive and exhaustive events, and a random variable X whose distri- bution depends on which of the 4,'s occurs (e.g., a com- muter might select one of three possible routes from home to work, with X representing the commute time). Let E(X4) denote the expected value of X given that the event occurs. Then it can can be shown that E(X) = E(X4) P(4) the weighted average of the indi- vidual "conditional expectations" where the weights are the probabilities of the partitioning events. A

a. The expected duration of a voice call to a particular telephone number is 3 minutes, whereas the expected duration of a data call to that same number is 1 minute. If 75% of all calls are voice calls, what is the expected. duration of the next call?

b. A deli sells three different types of chocolate chip cook- ies. The number of chocolate chips in a type i cookie has a Poisson distribution with parameter p=1+1 (i = 1, 2, 3). If 20% of all customers pur- chasing a chocolate chip cookie select the first type. 50% choose the second type, and the remaining 30% opt for the third type, what is the expected number of chips in a cookie purchased by the next customer?