Download Chapter 5: Continuous Probability Distribution

Chapter 5: Continuous Probability Distribution
Chapter 5: Continuous Probability Distribution
Keith E. Emmert
Department of Mathematics
Tarleton State University
June 16, 2011
Chapter 5: Continuous Probability Distribution
Outline
1
Why Do We Need to Know Models?
2
Modeling Continuous Variables
3
Normal Distributions
Chapter 5: Continuous Probability Distribution
Why Do We Need to Know Models?
Example
Scenario I: A patient visits his doctor complaining of a
number of symptoms. The doctor suspects the patient is
suffering from some disease. The doctor performs a diagnostic
test to check for this disease. High responses on the test
support that the patient may have the disease. The patients
test response is 200. What does this say?
Model for healthy
subjects
80
100
120
140
160
180
200
Test Response
Based on this model, it is very unlikely that a test response of
200, or greater, would have occurred if the subject were
actually healthy. Thus, either the patient has this disease or a
very unlikely event has occurred.
Chapter 5: Continuous Probability Distribution
Why Do We Need to Know Models?
Example
Scenario II: Suppose we wish to compare two drugs, Drug A
and Drug B, for relieving arthritis pain. Subjects suitable for
the study are randomized to one of the two drug groups and
are given instructions for dosage and how to measure their
“time to relief.” Results of the study are summarized by
presenting the models for the time to relief for the two drugs.
Drug A
Drug B
t
Time to relief
Consider any point in time, say time t as indicated on the
above axis. A higher proportion of subjects treated with Drug
A have felt “relief” by this time point as compared to those
treated with Drug B. If the study design was sound, then we
might conclude that Drug A works quicker than Drug B.
Chapter 5: Continuous Probability Distribution
Modeling Continuous Variables
Density Functions
A density function is a (nonnegative) function or curve that
describes the overall shape of a distribution. The total area under
the entire curve is equal to 1, and proportions or probabilities are
measured as areas under the density function.
As a simple example, below is a density curve (the blue curve).
The shaded area represents the probability that a random variable
takes on values between 6 and 20.
6
20
Chapter 5: Continuous Probability Distribution
Modeling Continuous Variables
Let’s Do It!
Using Density Functions
Density
Let the variable X represent the length of life, in years, for an
electrical component. The following figure is the density curve for
the distribution of X .
0.39
0
0.24
1
0.14
2
0.09
3
0.05
4
0.03
5
6
x
(a) What proportion of electrical components lasts longer than 6
years?
(b) What proportion of electrical components lasts longer than 1
year?
(c) Describe the shape of the distribution.
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Normal Distribution
The normal distribution has probability density function (or pdf)
given by
1
2
2
e −(x−µ) /2σ .
f (x) = √
2
2πσ
2
X is N(µ, σ ) means that the variable X is normally distributed
with mean
µ, variance σ 2 , and standard
deviation σ.
::::::: ::::::::::
:::::::::::::::::::
Below is an example of a typical normal curve. Notice it has
inflection points at x = µ ± σ and is symmetric about x = µ.
y
Μ-Σ
Μ
Μ+Σ
x
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Normal Distribution
Some Example Curves
NHΜ1 , Σ2 L
NHΜ2 , Σ1 L
NHΜ1 , Σ1 L
Μ1 , Μ1
Μ2
For N(µ1 , σ12 ) and N(µ2 , σ12 ), the mean is changed, which
slides the curve left or right. (Here µ2 > µ1 .)
For N(µ1 , σ12 ) and N(µ1 , σ22 ), the variance is changed, which
forces the curve to change it’s height (the distribution
becomes more or less “concentrated” as the variance
decreases or increases, respectively). (Here σ2 < σ1 .)
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Normal Distribution
Important Properties
1
A normal distribution is bell-shaped.
2
The mean, median, and mode are equal and are located at the
center of the distribution.
3
A normal distribution curve is unimodal (i.e. it has one mode).
4
The curve is symmetric about the mean.
5
The curve is continuous, that is, there are no gaps or holes.
For any x, there is a corresponding y .
6
The curve never touches the x-axis, but it does extend
infinitely in either direction.
7
The total area under a normal curve is 1. (Trust me.)
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Empirical Rule
For a random variable based upon the normal curve N(µ, σ 2 ), the
following empirical rule holds
Pr (µ − σ < X < µ + σ) ≈ 0.68: This means that the area
under the curve within 1 standard deviation of the mean is
approximately 68%.
Pr (µ − 2σ < X < µ + 2σ) ≈ 0.95: This means that the area
under the curve within 2 standard deviations of the mean is
approximately 95%.
Pr (µ − 3σ < X < µ + 3σ) ≈ 0.997: This means that the area
under the curve within 3 standard deviation of the mean is
approximately 99.7%.
Chapter 5: Continuous Probability Distribution
Normal Distributions
Example
Let the variable X represent IQ scores of 12-year-olds. Suppose
that the distribution of X is normal with a mean of 100 and a
standard deviation of 16that is, X is N(100, 162 ). Jessica is a
12-year-old and has an IQ score of 132.We would like to determine
the proportion of 12-year olds that have IQ scores less than
Jessicas score of 132. Since the area under the density curve
corresponds to proportion, we want to find the area to the left of
132 under an N(100, 162 ) curve. Sketch this curve and show the
corresponding area that represents this proportion.
100
132
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Normal
Standardizing Scores to Compare Observations from Different Normal Distributions
If X is N(µ, σ 2 ), then the standardized normal variable
X −µ
defined by Z =
is N(0, 1).
σ
Given a particular observation, x, of a random variable X that
is N(µ, σ 2 ), the z-score or standard score for an observed
value tells us how many standard deviations the observed
value is from the mean that is, it tells us how far the
observed value is from the mean in standard-deviation units.
It is calculated by:
x −µ
z=
.
σ
Chapter 5: Continuous Probability Distribution
Normal Distributions
Example
Comparing IQ Scores from Different Age Groups
Recall that the IQ scores for 12-year olds is N(100, 162 ).
Jessica had a score of 132. Compute Jessica’s standardized
score.
Suppose Jessica has an older brother, Mike, who is 20 years
old and has an IQ score of 144. It wouldnt make sense to
directly compare Mikes score of 144 to Jessicas score of 132.
The two scores come from different distributions due to the
age difference. Assume that the distribution of IQ scores for
20-year-olds is normal with a mean of 120 and a standard
deviation of 20, i.e. N(120, 202 ). Compute Mikes
standardized score.
Relative to their respective age group, who had the higher IQ
score - Jessica or Mike?
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Normal Distribution
Calculating Probability/Proportions
One way to calculate proportions is to standardize and then
use a table.
Another way is to use technology, such as the TI-84 or SPSS.
We’ll take this route.
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Normal Distribution
Calculating Probability/Proportions
As an example, using the standard normal, N(0, 1), we calculate
Pr (Z < 1.35) = 0.9114919. Notice that you do not type the mean
and standard deviation when using standard normal, the calculator
assumes it for you. You should type:
normalcdf(−E 99, 1.35) or normalcdf(−E 99, 1.35, 0, 1)
VARS
2nd
2
(-)
,
Opens the DISTR menu
Selects the
normalcdf(lower, upper, µ, σ) function
, 9 9 -E99 is minus
2nd
infinity
1
.
3
5
)
ENTER
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Normal Distribution
Calculating Probability/Proportions
As an example, using the standard normal, N(0, 1), we calculate
Pr (Z > 1.35) = 0.0885081. Type
normalcdf(1.35, E 99) or normalcdf(1.35, E 99, 0, 1)
2nd
2
,
2nd
VARS
Opens the DISTR menu
Selects the
normalcdf(lower, upper, µ, σ) function
E99 is plus in. 3 5
1
finity
,
9
9
)
ENTER
P(Z > 1.35) = 1 − Pr (Z ≤ 1.35) = 1 − Pr (Z < 1.35)
= 1 − 0.9114919 = 0.0885081.
Chapter 5: Continuous Probability Distribution
Normal Distributions
The Normal Distribution
Calculating Probability/Proportions
As an example, using a normal distribution, N(100, 162 ), we
calculate Pr (X < 132) = 0.9772499. Type
normalcdf(−E 99, 132, 100, 16)
VARS
2nd
2
(-)
Opens the DISTR menu
Selects the
normalcdf(lower, upper, µ, σ) function
, 9 9 -E99 is minus
2nd
infinity
,
1
,
100
3
2
,
16
)
ENTER
Chapter 5: Continuous Probability Distribution
Normal Distributions
Let’s Do It!
Standard Normal Areas
Find the area under the standard normal distribution between
z = 0 and z = 1.22. Sketch the area and use your calculator
to find the area.
Find the area under the standard normal distribution between
z = −1.22 and z = 1.22. Sketch the area and use your
calculator to find the area.
Find the area under the standard normal distribution to the
left of z = −1.22. Sketch the area and use your calculator to
find the area.
0
0
0
Chapter 5: Continuous Probability Distribution
Normal Distributions
Let’s Do It!
IQ Scores
We will continue with the model for IQ score of 12-year-olds.
Recall that X = IQ score of 12-year olds has a N(100, 162 )
distribution.
What proportion of the 12-year olds have IQ scores below 84?
What proportion of the 12-year olds have IQ scores 84 or
more?
What proportion of the 12-year olds have IQ scores between
84 and 116?
100
100
100
Chapter 5: Continuous Probability Distribution
Normal Distributions
Example
Top 1% of the IQ Distribution
Recall the N(100, 162 ) model for IQ scores of 12-year olds. What
IQ score must a 12-year old have to place in the top 1% of the
distribution of IQ scores? Type
invNorm(0.99, 100, 10)
VARS
2nd
3
100
Opens the DISTR menu
Selects the
invNorm(area, µ, σ) function
0
,
,
100
This yields an answer of 137.221566.
9
9
,
16
)
ENTER
Chapter 5: Continuous Probability Distribution
Normal Distributions
Let’s Do It!
Freestyle Swim Times
The finishing times for 11-12-year-old male swimmers performing
the 50-yard freestyle are normally distributed with a mean of 35
seconds and a standard deviation of 2 seconds.
(a) The sponsors of a swim meet decide to give certificates to all
11-12-year-old male swimmers who finish their 50-yard race in
under 32 seconds. If there are 50 such swimmers entered in
the 50-yard freestyle event, approximately how many
certificates will be needed?
(b) In what amount of time must a swimmer finish to be in the
“top” fastest 2% of the distribution of finishing times?
35
35
Chapter 5: Continuous Probability Distribution
Normal Distributions
Let’s Do It!
Hours Per Week
According to a study, men in the US devote an average of 16 hours
per week to house work. Assume that the number of hours men
devote to house work is normally distributed with a standard
deviation of 3.5.
(a) Suppose that the lower 10% of men on the distribution devote
fewer than x hours per week. Find the value of x.
(b) Suppose the upper 5% of men on the distribution devote more
than x hrs per week. Find the value of x.
16
16
Chapter 5: Continuous Probability Distribution
Normal Distributions
Let’s Do It!
Middle Portion of the Normal
If one-person household spends an average of $40 per month on
medications and doctor visits, find the maximum and minimum
dollar amounts spent per month for the middle 50% of one-person
household. Assume that the standard deviation is $5 and that the
amount spent is normally distributed.
40
Chapter 5: Continuous Probability Distribution
Normal Distributions
Homework
HW page 154: 1-14 all, 17, 18, 27, 28