Consider a collection A1, . . . , Ak of mutually exclusive and exhaustive events, and a random variable X whose distribution depends on which of the Ai’s occurs (e.g., a commuter might select one of three possible routes from home to work, with X representing the commute time). Let E(x|Ai ) denote the expected value of X given that the event Ai occurs. Then it can be shown that E(X) = ∑E(X|Ai ).P(Ai) the weighted average of the individual “conditional expectations” where the weights are the probabilities of the partitioning events.
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 cookies. The number of chocolate chips in a type i cookie has a Poisson distribution with parameter μi = i + 1 (i = 1, 2, 3). If 20% of all customers purchasing 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?
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