Question6. The duration of a telephone call (in seconds) has a uniform distribution over the interval . With probability , the telephone call is a roaming connection and a proportional charge of 2 cents per second is added; in other cases the telephone cost is included in a monthly standing charge of . There were 120 independent connections in May. a) Calculate the expected value of the invoice for May. b) Using the Poisson theorem, approximate the probability that at least three calls resulting in extra charges of at least each were made.
Studdy Solution
STEP 1
1. The duration of a call is uniformly distributed over .
2. The probability of a roaming connection is .
3. The charge for a roaming call is 2 cents per second.
4. The monthly standing charge is \$50.
5. There are 120 independent connections in May.
STEP 2
1. Calculate the expected value of the invoice for May.
2. Approximate the probability that at least three calls result in extra charges of at least \$1 each using the Poisson theorem.
STEP 3
Calculate the expected duration of a call. Since the duration is uniformly distributed over , the expected duration is:
STEP 4
Calculate the expected cost of a roaming call. The cost is 2 cents per second, so:
STEP 5
Calculate the expected cost of a single call. With probability , the call is roaming:
STEP 6
Calculate the expected total cost for 120 calls:
STEP 7
Add the monthly standing charge to the expected total cost:
STEP 8
Determine the condition for a call to result in extra charges of at least \$1. The duration must be at least 50 seconds, since:
STEP 9
Calculate the probability of a roaming call lasting at least 50 seconds. The probability of a call lasting at least 50 seconds is:
STEP 10
Calculate the probability of a roaming call lasting at least 50 seconds:
STEP 11
Use the Poisson approximation for the number of such calls. The expected number of such calls in 120 trials is:
STEP 12
Calculate the probability of at least three such calls using the Poisson distribution:
STEP 13
Calculate the individual probabilities:
For :
STEP 14
Calculate the cumulative probability:
STEP 15
Calculate the probability of at least three calls:
The expected value of the invoice for May is:
The probability that at least three calls resulted in extra charges of at least \$1 each is approximately:
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