Comcast Corporation is the largest cable television company,...

Comcast Corporation is the largest cable television company, the second largest Internet service provider, and the fourth largest telephone service provider in the United States. Generally known for quality and reliable service, the company periodically experiences unexpected service interruptions. On January $14,2009,$ such an interruption occurred for the Comcast customers living in southwest Florida. When customers called the Comcast office, a recorded message told them that the company was aware of the service outage and that it was anticipated that service would be restored in two hours. Assume that two hours is the mean time to do the repair and that the repair time has an exponential probability distribution. a. What is the probability that the cable service will be repaired in one hour or less? b. What is the probability that the repair will take between one hour and two hours? c. For a customer who calls the Comcast office at $1 : 00 \mathrm{PM},$ what is the probability that the cable service will not be repaired by $5 : 00 \mathrm{P.M.}$

Comcast Corporation is the largest cable television company, the second largest Internet
service provider, and the fourth largest telephone service provider in the United States.
Generally known for quality and reliable service, the company periodically experiences
unexpected service interruptions. On January $14,2009,$ such an interruption occurred for
the Comcast customers living in southwest Florida. When customers called the Comcast office, a recorded message told them that the company was aware of the service outage and
that it was anticipated that service would be restored in two hours. Assume that two hours
is the mean time to do the repair and that the repair time has an exponential probability
distribution.
a. What is the probability that the cable service will be repaired in one hour or less?
b. What is the probability that the repair will take between one hour and two hours?
c. For a customer who calls the Comcast office at $1 : 00 \mathrm{PM},$ what is the probability that
the cable service will not be repaired by $5 : 00 \mathrm{P.M.}$

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time until an event occurs, such as the length of time until a service is repaired or a component fails. It is characterized by a constant hazard rate, meaning that the probability of the event occurring in the next moment is independent of how much time has already elapsed.

Rate Parameter (?) and Mean

The rate parameter, often represented by ?, is the reciprocal of the mean of the exponential distribution. In practical terms, if the mean time until an event (such as a repair) is given, then ? is calculated as 1 divided by that mean. This parameter is fundamental to determining the probabilities associated with various time intervals.

Cumulative Distribution Function (CDF) and Probability Intervals

The cumulative distribution function (CDF) of the exponential distribution provides the probability that the event occurs within a certain time period. Probabilities over a specific interval can be calculated by taking the difference of the CDF evaluated at the endpoints of the interval. This approach is commonly used to determine the likelihood of an event happening within any given time frame.

Memoryless Property

One of the key features of the exponential distribution is its memoryless property, which means that the probability of an event occurring in the future is independent of how much time has already elapsed. This property is particularly useful in scenarios such as service interruptions or reliability studies, as it allows for the simplification of probability calculations regardless of prior elapsed time.

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