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6.5 Exercises

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This module provides a number of homework exercises related to Continuous Random Variables.

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.

  • What part of the experiment will yield discrete data?
  • What part of the experiment will yield continuous data?

When age is rounded to the nearest year, do the data stay continuous, or do they become discrete? Why?

Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a Uniform Distribution from 1 – 53 (spread of 52 weeks).

  • 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 a person is born at the exact moment week 19 starts. That is, find P ( X = 19 ) size 12{P \( X="19" \) } {} .
  • P ( 2 < X < 31 ) = size 12{P \( 2<X<"31" \) ={}} {}
  • Find the probability that a person is born after week 40.
  • {} P ( 12 < X ∣ X < 28 ) size 12{P \( "12"<X \lline X<"28" \) } {} =
  • Find the 70th percentile.
  • Find the minimum for the upper quarter.
  • X ~ U ( 1, 53 ) size 12{X " ~ " U \( 1,"53" \) } {}
  • f ( x ) = 1 52 size 12{f \( x \) = { {1} over { \( b - a \) } } = { {1} over { \( "53" - 1 \) } } = { {1} over {"52"} } } {} where 1 ≤ x ≤ 53 size 12{1<= x<= "53"} {}
  • 27
  • 15.01
  • 0
  • 29 52
  • 13 52
  • 16 27
  • 37.4
  • 40

A random number generator picks a number from 1 to 9 in a uniform manner.

  • X ~ size 12{X "~" } {}
  • Graph the probability distribution.
  • f ( x ) = size 12{f \( x \) ={}} {}
  • μ = size 12{μ={}} {}
  • σ = size 12{σ={}} {}
  • P ( 3 . 5 < X < 7 . 25 ) = size 12{P \( 3 "." 5<X<7 "." "25" \) ={}} {}
  • P ( X > 5 . 67 ) = size 12{P \( X>5 "." "67" \) ={}} {}
  • P ( X > 5 ∣ X > 3 ) = size 12{P \( X>5 \lline X>3 \) ={}} {}
  • Find the 90th percentile.

The speed of cars passing through the intersection of Blossom Hill Road and the Almaden Expressway varies from 10 to 35 mph and is uniformly distributed. None of the cars travel over 35 mph through the intersection.

  • 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={}} {}
  • What is the probability that the speed of a car is at most 30 mph?
  • What is the probability that the speed of a car is between 16 and 22 mph.
  • P ( 20 < X < 53 ) = size 12{P \( "20"<X<"53" \) ={}} {} State this in a probability question (similar to g and h ), draw the picture, and find the probability.
  • Find the 90th percentile. This means that 90% of the time, the speed is less than _____ mph while passing through the intersection per minute.
  • Find the 75th percentile. In a complete sentence, state what this means. (See j .)
  • Find the probability that the speed is more than 24 mph given (or knowing that) it is at least 15 mph.
  • X ~ U ( 10 , 35 ) size 12{X "~" U \( "10","35" \) } {}
  • f ( x ) = 1 25 where 10 ≤ X ≤ 35
  • 45 2
  • 7.22
  • 4 5
  • 6 25
  • 3 5
  • 32.5
  • 28.75
  • 11 20

According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between 6 and 15 pounds a month until they approach trim body weight. Let’s suppose that the weight loss is uniformly distributed. We are interested in the weight loss of a randomly selected individual following the program for one month. (Source: The McDougall Program for Maximum Weight Loss by John A. McDougall, M.D.)

  • 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 individual lost more than 10 pounds in a month.
  • Suppose it is known that the individual lost more than 10 pounds in a month. Find the probability that he lost less than 12 pounds in the month.
  • P ( 7 < X < 13 ∣ X > 9 ) = size 12{P \( 7<X<"13" \lline X>9 \) ={}} {} State this in a probability question (similar to g and h), draw the picture, and find the probability.

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.

  • 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

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.

  • 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.

Try these multiple choice problems

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

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

  • 2.0000
  • 2.4000
  • 2.750
  • 3.000

B

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

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Elementary statistics. OpenStax CNX. Dec 30, 2013 Download for free at http://cnx.org/content/col10966/1.4
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