Download Module 6: Continuous Probability Distributions

Module 6: Continuous
Probability Distributions
The Uniform Distribution
The uniform probability
distribution is perhaps
the simplest distribution
for a continuous random
variable.
This distribution is
rectangular in shape and
is defined by minimum
and maximum values.
The Uniform Distribution – Mean and
Standard Deviation
Southwest Arizona State University provides bus service to students while
they are on campus. A bus arrives at the North Main Street and
College Drive stop every 30 minutes between 6 A.M. and 11 P.M.
during weekdays. Students arrive at the bus stop at random times. The
time that a student waits is uniformly distributed from 0 to 30
minutes.
1.
2.
3.
4.
5.
Draw a graph of this distribution.
Show that the area of this uniform distribution is 1.00.
How long will a student “typically” have to wait for a bus? In other
words, what is the mean waiting time? What is the standard deviation
of the waiting times?
What is the probability a student will wait more than 25 minutes?
What is the probability a student will wait between 10 and 20
minutes?
The length of time patients must wait to see a doctor in a local clinic is uniformly
distributed between 15 minutes and 2 1/2 hours.
• What is the probability that a patient would have to wait between ¾ of an hour
and 2 hours?
• Calculate the probability that a patient would have to wait over 2 hours.
The travel time for a college student traveling between her home and her
college is uniformly distributed between 40 and 90 minutes.
• What is the probability that she will finish her trip in 80 minutes or
less?
• What is the probability that her trip will take longer than 60 minutes?
• What is the Standard Deviation of the distribution?
Characteristics of Normal Curve
Characteristics of Normal Curve
Areas Under the Curve
Applications of Normal Distribution
• A plant manager knows that the number of boxes of supplies
received weekly is normally distributed with a mean of 200
and a standard deviation of 20.
– What percentage of the time will the number of boxes
received weekly be greater than 210 or Less than 180?
– What percentage of the time will the number of boxes
received weekly be greater than 200
– What percentage of the time will the number of boxes
received weekly be less than 160?
What percentage of the time will the number of boxes
received weekly be greater than 210 or Less than 180?
– What percentage of the time will the number of boxes
received weekly be greater than 175
Normal Approximation to Binomial:
Continuity Correction Factor
The “Test” … 𝑛 𝑥 𝑝 > 5 𝐴𝑁𝐷 𝑛 𝑥 𝑞 > 5
Apply a Continuity Correction Factor of 0.50 to convert a Discrete
Variable to a Continuous Variable
Normal Approximation to the Binomial
• The probability that a call to an emergency help line is
answered in less than 10 seconds is 0.8. Assume that
the calls are independent of each other. Using the
normal approximation for binomial with a correction
for continuity, the probability that at least 75 of 100
calls are answered within 10 seconds is approximately
_______
• Assuming an equal chance of a new baby being a boy
or a girl we would like to find the probability of 40 or
more of the next 100 births at a local hospital will be
boys. Use the normal approximation for binomial with
a correction for continuity.
The “Magic Formula” for Normal
Distribution!
Solving for X
• Suppose you are working with a data set that
is normally distributed with a mean of 400
and a standard deviation of 20. Determine
the value of x such that 60% of the values are
greater than x.
• Suppose x is a normal random variable with
mean 60 and standard deviation 2. A z score
was calculated for a number, and the z score is
3.4. What is x?
Solving for σ and/or μ
• The weight of a product is normally distributed
with a mean 5 ounces. A randomly selected unit
of this product weighs 7.1 ounces. The probability
of a unit weighing more than 7.1 ounces is .0014.
The production supervisor has lost files
containing various pieces of information
regarding this process including the standard
deviation. Determine the value of standard
deviation for this process.
Solving for σ and/or μ
• If the mileage per gallon for a car is normally
distributed, 32 mpg has a z-score of 1.2, and 24
mpg has a z-score of -.4, what is the mean mpg of
the distribution?
• Given the length an athlete throws a hammer is a
normal random variable with a standard deviation
of 5 feet. If the probability he throws it no less
than 55 feet is 15.87%, what is the average
distance an athlete can throw a hammer?