# 5.6: Chapter Homework

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## 5.1 Properties of Continuous Probability Density Functions

*For each probability and percentile problem, draw the picture.*

Consider the following experiment. You are one of 100 people enlisted to take part in a study to determine the percent of nurses in America with an R.N. (registered nurse) degree. You ask nurses if they have an R.N. degree. The nurses answer “yes” or “no.” You then calculate the percentage of nurses with an R.N. degree. You give that percentage to your supervisor.

*For each probability and percentile problem, draw the picture.*72.Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks).

*Use the following information to answer the next three exercises.*The Sky Train from the terminal to the rental–car and long–term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution.**77**.What is the average waiting time (in minutes)?

*Use the following information to answer the next three exercises.*The average lifetime of a certain new cell phone is three years. The manufacturer will replace any cell phone failing within two years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution.**88**.The decay rate is:

- 0.3333
- 0.5000
- 2
- 3

What is the probability that a phone will fail within two years of the date of purchase?

- 0.8647
- 0.4866
- 0.2212
- 0.9997

**90**.What is the median lifetime of these phones (in years)?

- 0.1941
- 1.3863
- 2.0794
- 5.5452

**91**.At a 911 call center, calls come in at an average rate of one call every two minutes. Assume that the time that elapses from one call to the next has the exponential distribution.

- On average, how much time occurs between five consecutive calls?
- Find the probability that after a call is received, it takes more than three minutes for the next call to occur.
- Ninety-percent of all calls occur within how many minutes of the previous call?
- Suppose that two minutes have elapsed since the last call. Find the probability that the next call will occur within the next minute.
- Find the probability that less than 20 calls occur within an hour.

**92**.In major league baseball, a no-hitter is a game in which a pitcher, or pitchers, doesn't give up any hits throughout the game. No-hitters occur at a rate of about three per season. Assume that the duration of time between no-hitters is exponential.

- What is the probability that an entire season elapses with a single no-hitter?
- If an entire season elapses without any no-hitters, what is the probability that there are no no-hitters in the following season?
- What is the probability that there are more than 3 no-hitters in a single season?

**93**.During the years 1998–2012, a total of 29 earthquakes of magnitude greater than 6.5 have occurred in Papua New Guinea.

__Assume that the time spent waiting between earthquakes is exponential.__- What is the probability that the next earthquake occurs within the next three months?
- Given that six months has passed without an earthquake in Papua New Guinea, what is the probability that the next three months will be free of earthquakes?
- What is the probability of zero earthquakes occurring in 2014?
- What is the probability that at least two earthquakes will occur in 2014?

**94**.According to the American Red Cross, about one out of nine people in the U.S. have Type B blood. Suppose the blood types of people arriving at a blood drive are independent. In this case, the number of Type B blood types that arrive roughly follows the Poisson distribution.

- If 100 people arrive, how many on average would be expected to have Type B blood?
- What is the probability that over 10 people out of these 100 have type B blood?
- What is the probability that more than 20 people arrive before a person with type B blood is found?

**95**.A web site experiences traffic during normal working hours at a rate of 12 visits per hour. Assume that the duration between visits has the exponential distribution.

- Find the probability that the duration between two successive visits to the web site is more than ten minutes.
- The top 25% of durations between visits are at least how long?
- Suppose that 20 minutes have passed since the last visit to the web site. What is the probability that the next visit will occur within the next 5 minutes?
- Find the probability that less than 7 visits occur within a one-hour period.

**96**.At an urgent care facility, patients arrive at an average rate of one patient every seven minutes. Assume that the duration between arrivals is exponentially distributed.

- Find the probability that the time between two successive visits to the urgent care facility is less than 2 minutes.
- Find the probability that the time between two successive visits to the urgent care facility is more than 15 minutes.
- If 10 minutes have passed since the last arrival, what is the probability that the next person will arrive within the next five minutes?
- Find the probability that more than eight patients arrive during a half-hour period.