Transcript 03-w11-stats250-bgunderson-chapter-8-discrete

Author(s): Brenda Gunderson, Ph.D., 2011
License: Unless otherwise noted, this material is made available under the
terms of the Creative Commons Attribution–Non-commercial–Share
Alike 3.0 License: http://creativecommons.org/licenses/by-nc-sa/3.0/
We have reviewed this material in accordance with U.S. Copyright Law and have tried to maximize your
ability to use, share, and adapt it. The citation key on the following slide provides information about how you
may share and adapt this material.
Copyright holders of content included in this material should contact [email protected] with any
questions, corrections, or clarification regarding the use of content.
For more information about how to cite these materials visit http://open.umich.edu/education/about/terms-of-use.
Any medical information in this material is intended to inform and educate and is not a tool for self-diagnosis
or a replacement for medical evaluation, advice, diagnosis or treatment by a healthcare professional. Please
speak to your physician if you have questions about your medical condition.
Viewer discretion is advised: Some medical content is graphic and may not be suitable for all viewers.
Attribution Key
for more information see: http://open.umich.edu/wiki/AttributionPolicy
Use + Share + Adapt
{ Content the copyright holder, author, or law permits you to use, share and adapt. }
Public Domain – Government: Works that are produced by the U.S. Government. (17 USC § 105)
Public Domain – Expired: Works that are no longer protected due to an expired copyright term.
Public Domain – Self Dedicated: Works that a copyright holder has dedicated to the public domain.
Creative Commons – Zero Waiver
Creative Commons – Attribution License
Creative Commons – Attribution Share Alike License
Creative Commons – Attribution Noncommercial License
Creative Commons – Attribution Noncommercial Share Alike License
GNU – Free Documentation License
Make Your Own Assessment
{ Content Open.Michigan believes can be used, shared, and adapted because it is ineligible for copyright. }
Public Domain – Ineligible: Works that are ineligible for copyright protection in the U.S. (17 USC § 102(b)) *laws in
your jurisdiction may differ
{ Content Open.Michigan has used under a Fair Use determination. }
Fair Use: Use of works that is determined to be Fair consistent with the U.S. Copyright Act. (17 USC § 107) *laws in your
jurisdiction may differ
Our determination DOES NOT mean that all uses of this 3rd-party content are Fair Uses and we DO NOT guarantee that
your use of the content is Fair.
To use this content you should do your own independent analysis to determine whether or not your use will be Fair.
Chapter 8: Random Variables (page 43)
8.1 What is a Random Variable?
 Random variable represents the value
of the variable or characteristic of interest,
but before we look.
Definition:
 A random variable assigns a number to
each outcome of a random circumstance.
Equivalently, a random variable assigns a
number to each unit in a population.
Discrete versus Continuous
Definitions:

A discrete random variable can take one of a
countable list of distinct values.

A continuous random variable can take any
value in an interval or collection of intervals.
Try It! Discrete or Continuous?
A car is selected at random from used car dealership lot.
Decide whether characteristic is continuous or discrete.
a. Weight of the car (in pounds).
b. Number of seats (max passenger capacity).
c. Overall condition of car
(1 = good, 2 = very good, 3 = excellent).
d. Length of car (in feet).
Distribution of a Random Variable
Just what is the distribution of a random variable?

A model that shows what values are possible for
that random variable and how often those values
are expected to occur (i.e. their probabilities).

Expressed as a function or table or picture
(depending on its type).
Random Variable
Continuous
Discrete
Binomial
Uniform
Normal
More
to come
8.2 Discrete Random Variables



X = the random variable.
k = a number the discrete random variable could assume.
P(X = k) is probability the random variable X equals k.
Probability distribution function (pdf):
Value of X
Probability



p1
x2
p2
x3
p3
…
…
Two conditions are:


x1
Individual probabilities must be between 0 and 1.
Sum of all of individual probabilities must equal 1.
A probability histogram or stick graph.
Cumulative distribution function (cdf) is P(X  k).
Try It! Psychology Experiment
X = number of toys played with by children
X = # toys
0
1
2
3
4
Probability
0.03
0.16
0.30
0.23
0.17
5
.50
a. What is the missing
probability P(X = 5)?
.45
.40
.35
.30
.25
.20
Probability
b. Graph this
discrete pdf for X.
.15
.10
.05
0.00
0
X
1
2
3
4
5
c. Probability a child will play with at least 3 toys?
A)
B)
C)
D)
X = # toys
0
1
2
3
4
Probability
0.03
0.16
0.30
0.23
0.17
0.23
0.28
0.51
I don’t know
5
0.11
Try It! Psychology Experiment
X = number of toys played with by children
X = # toys
0
1
2
3
4
Probability
0.03
0.16
0.30
0.23
0.17
5
0.11
d. Given child has played with at least 3 toys, what is the
probability that he/she will play with all 5 toys?
Try It! Psychology Experiment
X = number of toys played with by children
X = # toys
0
1
2
3
4
Probability
0.03
0.16
0.30
0.23
0.17
5
0.11
e. Provide the cumulative distribution function of X.
X = # toys
0
1
2
Cum Prob P(X  k)
0.03
0.19
0.49
3
4
5
8.3 Expectations for Random Variables
pg 47
Definition:

The expected value of a random variable is the mean
value of the variable X in the sample space, or population,
of possible outcomes. Expected value, denoted by E(X).
Motivation …

Population = 100 families: 30 families have 1 child, 50 have
2 children, and 20 have 3 children. What is the mean
(average) number of children per family for this population?
2
1
2
1
2
2
1
2
2
3
3
Population of
100 families
etc.
Mean = (sum of all values)/100
= [1(30) + 2(50) + 3(20)]/100
= 1(30/100) + 2(50/100) + 3(20/100)
= 1(0.30) + 2(0.50) + 3(0.20)
= 1.9 children per family
Mean = Sum of (value x probability of that value)
Mean and Standard Deviation for Discrete
Definitions:
X is discrete random variable with values x1, x2, x3 …
occurring with probabilities p1, p2, p3…, then



Expected value (or mean) of X: m = E(X) =
Variance of X: V(X) =
s2
=
Standard deviation of X: s =

( xi  m ) 2 pi
2
(
x

m
)
pi
 i
 xi pi
Try It! Psychology Experiment
X = number of toys played with by children
X = # toys
0
1
2
3
4
5
Probability
0.03
0.16
0.30
0.23
0.17
0.11
a. What is the expected number of toys played with?
Try It! Psychology Experiment
X = number of toys played with by children
X = # toys
0
1
2
3
4
5
Probability
0.03
0.16
0.30
0.23
0.17
0.11
b. What is the standard deviation for the number of toys played with?
c.
Complete the interpretation of this standard deviation (in terms of
an average distance):
On average, the number of toys played with vary by about _______
from the mean number of toys played with of ________.
8.4 Binomial for Random Variables (pg 49)
Examples:

The number of girls in six independent births.

The number of tall men (over 6 feet) in a
random sample of 30 men from a large male
population.
Binomial Conditions
1. n “trials” where n determined in advance and is not
a random value.
2. Two possible outcomes on each trial, called
“success” and “failure” and denoted S and F.
3. Outcomes are independent from one trial to next.
4. Probability of “success” remains same from one
trial to next, denoted by p. Probability of a “failure”
is 1 – p for every trial.
A binomial random variable is defined as
X = number of successes in the n trials
of a binomial experiment.
Try It! Conditions Right for Binomial?
a. Observe gender of next 50 children born at a local hospital.
X = number of girls
b. Ten-question quiz has five T-F questions, and five
multiple-choice questions each with four possible choices.
Student randomly picks an answer for every question.
X = number of answers that are correct
Try It! Conditions Right for Binomial?
c. Four students are randomly picked without replacement from
large student body listing of 1000 women and 1000 men.
X = number of women among the four selected students.
What if student body listing was of 10 women and 10 men?
Rule of Thumb: popul at least 10 times as large as sample  ok!
The Binomial Formula: Shoppers
Suppose 25% of online shoppers actually make a purchase and
have a random sample of 10 such shoppers. If stated rate true,
what is the probability that ...
... all 10 shoppers will actually make a purchase?
...none of the shoppers will make a purchase?
The Binomial Formula: Shoppers
Suppose 25% of online shoppers actually make a purchase and
have a random sample of 10 such shoppers. If stated rate true,
what is the probability that ...
... just 1 shopper will actually make a purchase?
The Binomial Distribution
n k
nk
P( X  k )    p (1  p )
k 
for k = 0, 1, 2, …, n
where
n
n!
  
 k  k! (n  k )!
which represents the number of ways to select k items from n
Try It! The first part…
1.
10 
  
0
2.
10 
  
10 
3.
10 
  
1
4.
10 
  
9
5.
10 
  
2