Download Section 6.3 geometric distribution

Section 6.3 Geometric Distribution
Statistics
In the last section we looked at the probability that a certain number of people
in a group will have a certain characteristic. For example, in a group of 15
Congolese citizens picked at random, we could determine that exactly 3 of them
would live in Kinshasa.
Geometric distributions look at a different type of probability. An example of a
geometric distribution type of probability would be if I were to interview
Congolese citizens picked at random, what is the probability that the third
person would be the first person I interviewed from Kinshasa?
Geometric distribution problems are sometimes called ____________________
problems because you are waiting until the first event of something happens.
These problems also differ from binomial distribution problems because the
number of ______________ is not fixed. The number of trials for each
situation is the number of times it takes until the first occurrence of the event of
interest.
What reasoning can we use to determine the probability for our first “success”
to be a certain trial number? One condition that is the same as for binomial
distributions is that we have two possible outcomes, ___________________ or
__________________. If the probability of success is _______, the probability
of failure will be ____________ (let us call that q for now). Using p and q as
probabilities for “success” or “failure”, then, the probability for our first success
being on the fourth trial, say, would be:
P X  4  _______  _______  _______  _______
As with the binomial distribution, there are four conditions that must be met for
a geometric distribution:
1.) They are ________________. (Each trial will have either a _________
or a __________________.
2.) Each trial is __________________ of the others (the result of one trial
has no ______________ on another trial).
3.) The trials continue until the first _______________ occurs.
4.) The probability, ______, of success is the ____________ for each
trial, ______ < ______ < ______.
The distribution of the random variable X that counts the number of trials
needed until the first “success” occurs is called a ________________________.
The probability that the first success occurs on the X = ________ trial is:
______________________________________________
for k = 1, 2, 3, . . .
The formulas for the Expected number of trials before the first success and the
standard deviation of the expected number are:
______________________
and
_____________________
If you are interested in the number of trials before the nth success occurs, you
just multiply n by the expected value:
Expected number of trials before the nth success = __________________
The TI calculators have a built in function for computing binomial probabilities.
The first is geometpdf(p, k). It is found by doing 2nd VARS, under DIST go
down to D. Typing in p and k and ENTER gives the probability that the first
occurrence will be on the kth trial. If k is entered in { brackets (2nd”(“ and 2nd
“)”) you can enter any number of k values separated by “,” and get probability
returns for all of those k values.
The second is geometcdf(p,k). This is found by doing 2nd VARS, under DIST
go down to E. Typing in p and k and ENTER gives the cumulative probability
that the first occurrence will be on the 1st, 2nd, 3rd, ... ,to kth trial.