Transcript 8-2 geometric distribution

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 8:
The Binomial and
Geometric Distributions
Copyright © 2008 by W. H. Freeman & Company
Geometric Setting
• Flip a coin until you get a head.
• Roll a die until you get a 3.
• In basketball, attempt a 3-point shot until
you make a basket.
• All involve counting the number of trials
until an event of interest happens.
Geometric Distribution
• Distribution produced by a geometric
random variable.
Geometric Setting?
We want to draw cards without replacement
from a deck of 52 until you draw an ace.
Is this a geometric setting?
•Binary?
•Independent?
•Constant p for every trial?
•Number of trials is the variable of interest?
Geometric Setting?
We want to roll a single die until we get a
three.
Is this a geometric setting?
•Binary?
•Independent?
•Constant p for every trial?
•Number of trials is the variable of interest?
Geometric Probability
We want to roll a single die until we get a
three.
What is the probability of getting a 3 on the
first roll?
What is the probability of getting a 3 on the
second roll?
What is the probability of getting a 3 on the
third roll?
Geometric Sequence
Note that while a geometric sequence is
infinite, the SUM of the geometric
probabilities converge to 1 which is necessary
to have a valid probability distribution.
See p. 542 for a more detailed explanation.
Geometric Probability Distribution
Table
X= number of rolls of a die until a 3 occurs.
X:
1
2
3
4
5…
P(X): 0.1667 0.1389 0.1157 0.0965 0.0804 …
Geometric distributions are always skewed right.
Notice that the second formula is for variance not standard deviation.
Also: Neither formula is on the magic sheet. You must memorize.
And no, I don’t know why. 
Mean, Variance, and Standard
Deviation
Ex. 8.18 Arcade Game
X= number of tosses needed to land a coin in a
saucer. p =1/12, tosses are independent
What is the expected number of tosses required to
win?
What is the variance for the distribution?
What is the standard deviation for the distribution?
P(X > n) = 1 – P(X ≤ n)
You can use geomcdf(p, n) to calculate P(X ≤ n).
Note that these formulas are also NOT on the magic sheet!!
Fancy derivation on P.546. Enjoy!
P(X > n)
Ex. 8.19 Arcade Game
X= number of tosses needed to land a coin in a
saucer. p =1/12, tosses are independent
What is the probability that it takes more than 12
tosses to win?
What is the probability that it takes more than 24
tosses to win?
Wait-time Simulation
1. State the problem or describe the random
phenomenon. State stopping rule.
2. State the assumptions. (individual likeliness,
independent trials)
3. Assign digits to represent outcomes. Are
repeated digits okay?
4. Simulate many repetitions.
5. State your conclusions.
Wait-time Simulation
1.
2.
3.
4.
5.
Determine the number of boxes of Cheerios you would
expect to buy in order to get one of the “free” dollar bills.
Stop when you get a box with a dollar bill.
Each box is equally likely to have a dollar bill (p = 1/20).
The contents of each box are independent of each other.
Let 00 to 04 represent boxes that contain a dollar bill. Let 05
to 99 represent boxes that do not contain a dollar bill. Peel
off pairs of numbers until you get a pair that represents a box
containing a dollar bill.
Simulate many repetitions.
State your conclusion.