5.3 The Exponential Distribution (Optional)

c. Which is larger, the mean or the median?

The number of days ahead travelers purchase their airline tickets can be modeled by an exponential distribution with the average amount of time equal to 15 days. Find the probability that a traveler will purchase a ticket fewer than 10 days in advance. How many days do half of all travelers wait?

On the average, a certain computer part lasts 10 years. The length of time the computer part lasts is exponentially distributed.

a. What is the probability that a computer part lasts more than seven years?

b. On the average, how long would five computer parts last if they are used one after another?

On average, a pair of running shoes can last 18 months if used every day. The length of time running shoes last is exponentially distributed. What is the probability that a pair of running shoes last more than 15 months? On average, how long would six pairs of running shoes last if they are used one after the other? Eighty percent of running shoes last at most how long if used every day?

Suppose that the length of a phone call, in minutes, is an exponential random variable with decay parameter $\frac{1}{12}$. If another person arrives at a public telephone just before you, find the probability that you will have to wait more than five minutes. Let X = the length of a phone call, in minutes.

What is m, μ, and σ? The probability that you must wait more than five minutes is _______ .

Suppose that the distance, in miles, that people are willing to commute to work is an exponential random variable with a decay parameter $\frac{1}{20}$. Let X = the distance people are willing to commute in miles. What is m, μ, and σ? What is the probability that a person is willing to commute more than 25 miles?

Suppose that the longevity of a light bulb is exponential with a mean lifetime of eight years. If a bulb has already lasted 12 years, find the probability that it will last a total of more than 19 years.

In a small city, the number of automobile accidents occur with a Poisson distribution at an average of three per week.

1. Calculate the probability that at most two accidents occur in any given week.
2. What is the probability that there are at least two weeks between any two accidents?

Use the following information to answer the next 10 exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)

Are outcomes equally likely in this distribution? Why or why not?

Use the following information to answer the next 10 exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)

What is the mean?

Use the following information to answer the next 10 exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)

State the probability density function.

Use the following information to answer the next 10 exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)

Find P(2 < x < 10).

Use the following information to answer the next 10 exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)

Find the 70^th percentile.

Use the following information to answer the next eight exercises. A distribution is given as X ~ Exp(0.75).

What is the probability density function?

Use the following information to answer the next eight exercises. A distribution is given as X ~ Exp(0.75).

Find the 30^th percentile.

Use the following information to answer the next eight exercises. A distribution is given as X ~ Exp(0.75).

Which is larger, the mean or the median?

Use the following information to answer the next eight exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

Are the data discrete or continuous?

Use the following information to answer the next eight exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

What is the decay rate (m)?

Suppose that the useful life of a particular car battery, measured in months, decays with parameter 0.025. We are interested in the life of the battery.

1. Define the random variable. X = _________________________________.
2. Is X continuous or discrete?
3. X ~ ________
4. On average, how long would you expect one car battery to last?
5. On average, how long would you expect nine car batteries to last, if they…

The time (in years) after reaching age 60 that it takes an individual to retire is approximately exponentially distributed with a mean of about five years. Suppose we randomly pick one retired individual. We are interested in the time after age 60 to retirement.

1. Define the random variable. X = _________________________________.
2. Is X continuous or discrete?
3. X ~ = ________
4. μ = ________
5. σ = ________
6. …

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.

What is the decay rate?

1. 0.3333
2. 0.5000
3. 2
4. 3

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.

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

1. 0.1941
2. 1.3863
3. 2.0794
4. 5.5452

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.

In major league baseball, a no-hitter is a game in which a pitcher, or pitchers, doesn't giv…

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.

According to the American Red Cross, about one out of nine people in the United States have…

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.

At an urgent care facility, patients arrive at an average rate of one patient every seven mi…