Transcript Geometric_Distribution

Geometric Distribution
The geometric distribution computes the probability that the
first success of a Bernoulli trial comes on the xth trial.
For example, suppose we want to roll a four on a six-sided
die. We will roll until we get a four, then we stop.
The random variable is the number of rolls until we get a
four.
The probability of getting a four on the first roll is P(1)=1/6.
The probability of getting a four for the first time on the
second roll is P(2)=(5/6)(1/6)=5/36.
The probability of getting the first four on the third roll is
P(3)=(5/6)(5/6)(1/6)=25/216.
In general, we see that all but the last roll are not fours and so
have a probability of 5/6 and the last roll will have a probability
of 1/6.
So, P(X)=(5/6)(X-1)(1/6)
Generalizing to other problems, it is not so difficult to see that
P(X )  (1  p)
X1
p
as there are X-1 failures followed by one success.
We can work many problems on the TI-83, although the
formula is so simple that you may just want to work them
directly.
Problem: Compute the probability that the first four occurs on
the third roll of the die.
On the calculator, press <2nd> <DISTR> <D:geometpdf(>
then type in 1/6,3). (You can convert to a fraction if you like
with <MATH> <1:Frac> <ENTER>.)
As you see, we get the same answer as we got earlier.
We can use a sequence to get several probabilities at once.
Problem: Compute the probabilities that the first four occurs
on the first through fifth rolls of a fair die.
This example will be worked on the homescreen so that
we can see fractions. Most often I work in lists, as we did
with the binomial distribution.
Press: <2nd> <DISTR> <D:geometpdf(> then type
1/6,{1,2,3,4,5}) <ENTER>.
Now press: <MATH> <1:Frac> <ENTER>
Hint: The { symbol
uses the 2nd
function, then the (
key.
Use the toggle arrows to scroll
to see the other answers.
{1/6, 5/6, 25/216, 125/1296, 625/7776}
Problem: Find the probability of getting a four in the first
three trials, i.e., P(X3).
In this case we use the cumulative density function geometcdf.
On the TI-83, press: <2nd> <DISTR>
<E:geometcdf(> then type 1/6,3)
As you see, we find the probability is
91/216.
We can also make relative frequency histograms for the pdf
and the cdf.
Problem: Construct relative frequency histograms for the first
four on rolls of a six-sided die. Show X=1 through X=7.
Enter 1 through 7 in L1. Let L2 = geometpdf(1/6,L1). Set a
window as shown, and set up the STAT PLOT.
Trace to see the area of each column in the histogram.
Last Problem: Plot the histogram for the cumulative density
function for the same experiment. Show X=1 through X=7.
Enter 1 through 7 in L1. Let L2 = geometcdf(1/6,L1). Set a
window as shown, and set up the STAT PLOT.
Trace to see the area of each column in the histogram.
The end