Transcript 8.2.2 - GEOCITIES.ws

The Expected Value of
Geometric Distributions
Section 8.2.2
Starter 8.2.2
• The SAT Math and Verbal sections are
both designed to have a mean of 500 and
standard deviation of 100.
• If we defined a new measure (TOTAL) by
adding the scores on the two sections,
what would you expect the mean and
standard deviation of TOTAL to be?
Objectives
• Calculate the expected value (mean) of a
geometric random variable.
California Standards
5.0 Students know the definition of the mean of a
discrete random variable and can determine the mean
for a particular discrete random variable.
7.0 Students demonstrate an understanding of the
standard distributions (normal, binomial, and
exponential) and can use the distributions to solve for
events in problems in which the distribution belongs to
those families.
Activity
• Craps players are interested in how many rolls
they should expect on average before a 7
comes up. What is your guess of the answer?
Write it down.
• Roll a pair of dice until you get a 7. Write down
how many rolls it took to get the 7.
• Repeat until you have a total of 10 trials. Report
your results to me when I call for them. I will
record them and display the distribution the
class got.
• Now answer the question based on our
experimental distribution.
Calculating the Expected Value
• Here is the PDF of a general geometric
distribution. Recall from Chapter 7 (page 387)
how we found the mean of any random variable.
Write an expression for the mean in this case.
X
1
2
3
4
…
P(X)
p
pq
pq2
pq3
…
  1( p)  2( pq)  3( pq )  4( pq )  ...
2
3
 p(1  2q  3q  4q  ...)
2


1
 p
2 
 1  2q  q 
3
(Not obvious; prove inductively)
 1 
 p
2
 1  q  
 1 
 p 2 
p 
1

p
So, for example, if the probability
1
of rolling a 7 is , then it should
6
take about 6 rolls to get a 7.
Objectives
• Calculate the expected value (mean) of a
geometric random variable.
California Standards
5.0 Students know the definition of the mean of a
discrete random variable and can determine the mean
for a particular discrete random variable.
7.0 Students demonstrate an understanding of the
standard distributions (normal, binomial, and
exponential) and can use the distributions to solve for
events in problems in which the distribution belongs to
those families.
Homework
• Read pages 441 – 443
• Do problems 27 – 29