Consider a collection A1, … , Ak of mutually exclusive andexhaustive events, and a random variable X whose distributiondepends on which of the Ai’s occurs (e.g., a commutermight select one of three possible routes from hometo work, with X representing the commute time). LetE(XuAi) denote the expected value of X given that the eventAi occurs. Then it can be shown that E(X) 5oE(XuAi) ? P(Ai), the weighted average of the individual“conditional expectations” where the weights are the probabilitiesof the partitioning events.a. The expected duration of a voice call to a particulartelephone 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 expectedduration of the next call?b. A deli sells three different types of chocolate chipcookies. The number of chocolate chips in a type icookie has a Poisson distribution with parametermi 5 i 1 1 (i 5 1, 2, 3). If 20% of all customerspurchasing a chocolate chip cookie select the firsttype, 50% choose the second type, and the remaining30% opt for the third type, what is the expected numberof chips in a cookie purchased by the next customer?
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(XuAi) denote the expected value of X given that the
Ai occurs. Then it can be shown that E(X) 5
oE(XuAi) ? 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
mi 5 i 1 1 (i 5 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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