Download Section 6.3 Third Day Geometric RVs

Section 6.3
Geometric Random Variables

Geometric Settings
Definition:
A geometric setting arises when we perform independent trials of the
same chance process and record the number of trials until a particular
outcome occurs. The four conditions for a geometric setting are
B
• Binary? The possible outcomes of each trial can be classified as
“success” or “failure.”
I
• Independent? Trials must be independent; that is, knowing the result
of one trial must not have any effect on the result of any other trial.
T
• Trials? The goal is to count the number of trials until the first success
occurs.
S
• Success? On each trial, the probability p of success must be the
same.
Binomial and Geometric Random Variables
In a binomial setting, the number of trials n is fixed and the binomial random
variable X counts the number of successes. In other situations, the goal is to
repeat a chance behavior until a success occurs. These situations are called
geometric settings.
Comparison of Binomial to
Geometric
Binomial
Geometric
Each observation has two outcomes
(success or failure).
Each observation has two outcomes
(success or failure).
The probability of success is the
same for each observation.
The probability of success is the
same for each observation.
The observations are all
independent.
The observations are all
independent.
There are a fixed number of trials.
There is a fixed number of
successes (1).
So, the random variable is how
many successes you get in n trials.
So, the random variable is how
many trials it takes to get one
success.
How to Calculate Geometric
Probabilities

It’s usually not difficult to calculate these
by hand.


Let’s revisit Chris’ free throw shooting.
Currently, he is a 75% free throw shooter.

What is the P(X = 3)? That means what is the
probability that he makes his first basket on the
third shot?
Let’s Construct a Probability Distribution for
a Geometric Random Variable



Suppose Coach Roth helps Chris improve
to be a 80% free throw shooter.
Begin constructing a probability
distribution for how many shots it takes
for him to make his first free throw.
What would the graph of a geometric
probability distribution look like?
The Probability Distribution for the
Geometric R.V.

Calculating a > or a ≥ probability should
use the converse rule.


So, P(X > 6) = 1 – P(X ≤ 6)
Using the “New Chris” example, find the
following probabilities.

P(X<3)
P(X≥5)
Mean & Variance of Geometric RV

The formula for the mean of a geometric RV is
1
X 
p

The formula for the variance of a geometric RV is
(1  p)
 
p2
2
X

These formulas are NOT given to you on the
exam.
Mean & Variance of Geometric RV

Let’s revisit “New Chris” one more time.

Let X = when Chris ______________

What is the expected value of X?

What is the standard deviation of X?
Putting it all together…
We’ve studied two large categories of RVs:
discrete and continuous


Among the discrete RVs, we’ve studied the binomial and geometric
 The graph of a binomial RV can be skewed left, symmetric,
or skewed right, depending on the value of p.
 The graph of a geometric RV is ALWAYS skewed right.
Always.
 Other discrete RVs can be given to you in the form of a
table.
Among the continuous, we’ve studied the normal RVs.
 To find probabilities of a normal RV, convert to a Z score and
use Table A.