5.7 Homework  (Page 2/2)

A subway train on the Red Line arrives every 8 minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution.

• a Define the random variable. $X=$
• b $X\text{~}$
• c Graph the probability distribution.
• d $f(x)=$
• e $\mu =$
• f $\sigma =$
• g Find the probability that the commuter waits less than one minute.
• h Find the probability that the commuter waits between three and four minutes.
• i 60% of commuters wait more than how long for the train? State this in a probability question (similar to g and h ), draw the picture, and find the probability.

The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. We randomly select one first grader from the class.

• a Define the random variable. $X=$
• b $X\text{~}$
• c Graph the probability distribution.
• d $f(x)=$
• e $\mu =$
• f $\sigma =$
• g Find the probability that she is over 6.5 years.
• h Find the probability that she is between 4 and 6 years.
• i Find the 70th percentile for the age of first graders on September 1 at Garden Elementary School.

Let $X\text{~}$ Exp(0.1)

• a decay rate=
• b $\mu =$
• c Graph the probability distribution function.
• d On the above graph, shade the area corresponding to and find the probability.
• e Sketch a new graph, shade the area corresponding to and find the probability.
• f Sketch a new graph, shade the area corresponding to $P(x>7)$ and find the probability.
• g Sketch a new graph, shade the area corresponding to the 40th percentile and find the value.
• h Find the average value of $x$ .

Suppose that the length of long distance phone calls, measured in minutes, is known to have an exponential distribution with the average length of a call equal to 8 minutes.

• a Define the random variable. $X=$
• b Is $X$ continuous or discrete?
• c $X\text{~}$
• d $\mu =$
• e $\sigma =$
• f Draw a graph of the probability distribution. Label the axes.
• g Find the probability that a phone call lasts less than 9 minutes.
• h Find the probability that a phone call lasts more than 9 minutes.
• i Find the probability that a phone call lasts between 7 and 9 minutes.
• j If 25 phone calls are made one after another, on average, what would you expect the total to be? Why?

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.

• a Define the random variable. $X=$
• b Is $X$ continuous or discrete?
• c $X\text{~}$
• d On average, how long would you expect 1 car battery to last?
• e On average, how long would you expect 9 car batteries to last, if they are used one after another?
• f Find the probability that a car battery lasts more than 36 months.
• g 70% of the batteries last at least how long?

The percent of persons (ages 5 and older) in each state who speak a language at home other than English is approximately exponentially distributed with a mean of 9.848 . Suppose we randomly pick a state. (Source: Bureau of the Census, U.S. Dept. of Commerce)

• a Define the random variable. $X=$
• b Is $X$ continuous or discrete?
• c $X\text{~}$
• d $\mu =$
• e $\sigma =$
• f Draw a graph of the probability distribution. Label the axes.
• g Find the probability that the percent is less than 12.
• h Find the probability that the percent is between 8 and 14.
• i The percent of all individuals living in the United States who speak a language at home other than English is 13.8 .
  • i Why is this number different from 9.848%?
  • ii What would make this number higher than 9.848%?

The time (in years) after reaching age 60 that it takes an individual to retire is approximately exponentially distributed with a mean of about 5 years. Suppose we randomly pick one retired individual. We are interested in the time after age 60 to retirement.

• a Define the random variable. $X=$
• b Is $X$ continuous or discrete?
• c $X\text{~}$
• d $\mu =$
• e $\sigma =$
• f Draw a graph of the probability distribution. Label the axes.
• g Find the probability that the person retired after age 70.
• h Do more people retire before age 65 or after age 65?
• i In a room of 1000 people over age 80, how many do you expect will NOT have retired yet?

The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of $150.

• a Define the random variable. $X=$
• b $X\text{~}$
• c $\mu =$
• d $\sigma =$
• e Draw a graph of the probability distribution. Label the axes.
• f Find the probability that a car required over $300 for maintenance during its first year.

The decay rate is

• A 0.3333
• B 0.5000
• C 2.0000
• D 3.0000

What is the probability that a phone will fail within 2 years of the date of purchase?

• A 0.8647
• B 0.4866
• C 0.2212
• d 0.9997

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

• A 0.1941
• B 1.3863
• C 2.0794
• D 5.5452

What is the average waiting time (in minutes)?

• A 0.0000
• B 2.0000
• C 3.0000
• D 4.0000

Find the 30th percentile for the waiting times (in minutes).

• A 2.0000
• B 2.4000
• C 2.750
• D 3.000

The probability of waiting more than 7 minutes given a person has waited more than 4 minutes is?

• A 0.1250
• B 0.2500
• C 0.5000
• D 0.7500