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 ?i = 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?
Answer Step 1 of 4 Given E(X) = E(X|Ai ).P(Ai) a)The expected duration of a voice call to a particular telephone number is 3 minutes E(X|A1)=3, P(A1)=0.75 The expected duration of a data call to a particular telephone number is 1 minutes E(X|A1)=1, P(A2)=0.25 Step 2 of 4 Given E(X) = E(X|Ai ).P(Ai) =3(0.75)+1(0.25) =2.5 The expected duration of the next call is 2.5 minutes