<< Chapter < Page Chapter >> Page >

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.

  • Define the random variable. X = size 12{X={}} {}
  • X ~ size 12{X "~" } {}
  • Graph the probability distribution.
  • f ( x ) = size 12{f \( x \) ={}} {}
  • μ = size 12{μ={}} {}
  • σ = size 12{σ={}} {}
  • Find the probability that the commuter waits less than one minute.
  • Find the probability that the commuter waits between three and four minutes.
  • 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.
  • X ~ U ( 0,8 ) size 12{X "~" U \( 0,8 \) } {}
  • f ( x ) = 1 8 where 0 x 8
  • 4
  • 2.31
  • 1 8
  • 1 8
  • 3.2
Got questions? Get instant answers now!

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.

  • Define the random variable. X = size 12{X={}} {}
  • X ~ size 12{X "~" } {}
  • Graph the probability distribution.
  • f ( x ) = size 12{f \( x \) ={}} {}
  • μ = size 12{μ={}} {}
  • σ = size 12{σ={}} {}
  • Find the probability that she is over 6.5 years.
  • Find the probability that she is between 4 and 6 years.
  • Find the 70th percentile for the age of first graders on September 1 at Garden Elementary School.
Got questions? Get instant answers now!

Let X ~ size 12{X "~" } {} Exp(0.1)

  • decay rate=
  • μ = size 12{μ={}} {}
  • Graph the probability distribution function.
  • On the above graph, shade the area corresponding to P ( x < 6 ) size 12{P \( X<6 \) } {} and find the probability.
  • Sketch a new graph, shade the area corresponding to P ( 3 < x < 6 ) size 12{P \( 3<X<6 \) } {} and find the probability.
  • Sketch a new graph, shade the area corresponding to P ( x > 7 ) size 12{P \( X>7 \) } {} and find the probability.
  • Sketch a new graph, shade the area corresponding to the 40th percentile and find the value.
  • Find the average value of x size 12{X} {} .
  • 0.1
  • 10
  • 0.4512
  • 0.1920
  • 0.4966
  • 5.11
  • 10
Got questions? Get instant answers now!

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.

  • Define the random variable. X = size 12{X={}} {}
  • Is X size 12{X} {} continuous or discrete?
  • X ~ size 12{X "~" } {}
  • μ = size 12{μ={}} {}
  • σ = size 12{σ={}} {}
  • Draw a graph of the probability distribution. Label the axes.
  • Find the probability that a phone call lasts less than 9 minutes.
  • Find the probability that a phone call lasts more than 9 minutes.
  • Find the probability that a phone call lasts between 7 and 9 minutes.
  • If 25 phone calls are made one after another, on average, what would you expect the total to be? Why?
Got questions? Get instant answers now!

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.

  • Define the random variable. X = size 12{X={}} {}
  • Is X size 12{X} {} continuous or discrete?
  • X ~ size 12{X "~" } {}
  • On average, how long would you expect 1 car battery to last?
  • On average, how long would you expect 9 car batteries to last, if they are used one after another?
  • Find the probability that a car battery lasts more than 36 months.
  • 70% of the batteries last at least how long?
  • X ~ Exp ( 0.025 ) size 12{X "~" "Exp" \( { {1} over {5} } \) } {}
  • 40 months
  • 360 months
  • 0.4066
  • 14.27
Got questions? Get instant answers now!

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)

  • Define the random variable. X = size 12{X={}} {}
  • Is X size 12{X} {} continuous or discrete?
  • X ~ size 12{X "~" } {}
  • μ = size 12{μ={}} {}
  • σ = size 12{σ={}} {}
  • Draw a graph of the probability distribution. Label the axes.
  • Find the probability that the percent is less than 12.
  • Find the probability that the percent is between 8 and 14.
  • The percent of all individuals living in the United States who speak a language at home other than English is 13.8 .
    • Why is this number different from 9.848%?
    • What would make this number higher than 9.848%?
Got questions? Get instant answers now!

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.

  • Define the random variable. X = size 12{X={}} {}
  • Is X size 12{X} {} continuous or discrete?
  • X ~ size 12{X "~" } {}
  • μ = size 12{μ={}} {}
  • σ = size 12{σ={}} {}
  • Draw a graph of the probability distribution. Label the axes.
  • Find the probability that the person retired after age 70.
  • Do more people retire before age 65 or after age 65?
  • In a room of 1000 people over age 80, how many do you expect will NOT have retired yet?
  • X ~ Exp ( 1 5 ) size 12{X "~" "Exp" \( { {1} over {5} } \) } {}
  • 5
  • 5
  • 0.1353
  • Before
  • 18.3
Got questions? Get instant answers now!

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

  • Define the random variable. X = size 12{σ={}} {}
  • X ~ size 12{X "~" } {}
  • μ = size 12{μ={}} {}
  • σ = size 12{σ={}} {}
  • Draw a graph of the probability distribution. Label the axes.
  • Find the probability that a car required over $300 for maintenance during its first year.
Got questions? Get instant answers now!

Try these multiple choice problems

The next three questions refer to the following information. The average lifetime of a certain new cell phone is 3 years. The manufacturer will replace any cell phone failing within 2 years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution.

The decay rate is

  • 0.3333
  • 0.5000
  • 2.0000
  • 3.0000

A

Got questions? Get instant answers now!

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

  • 0.8647
  • 0.4866
  • 0.2212
  • 0.9997

B

Got questions? Get instant answers now!

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

  • 0.1941
  • 1.3863
  • 2.0794
  • 5.5452

C

Got questions? Get instant answers now!

The next three questions refer to the following information. The Sky Train from the terminal to the rental car and long term parking center is supposed to arrive every 8 minutes. The waiting times for the train are known to follow a uniform distribution.

What is the average waiting time (in minutes)?

  • 0.0000
  • 2.0000
  • 3.0000
  • 4.0000

D

Got questions? Get instant answers now!

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

  • 2.0000
  • 2.4000
  • 2.750
  • 3.000

B

Got questions? Get instant answers now!

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

  • 0.1250
  • 0.2500
  • 0.5000
  • 0.7500

B

Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics' conversation and receive update notifications?

Ask