I can prove anything by statistics - except the
Having spent much time studying binomial distributions, what qualifies, formulas to use, etc., learning about geometric distributions should be easy.
Let's start by contrasting the two distributions and settings.
Binomial: has a FIXED number of trials before the experiment begins and X counts the number of successes obtained in that fixed number.
Geometric: has a fixed number of successes (ONE...the FIRST) and counts the number of trials needed to obtain that first success. It is theoretically possible to proceed indefinitely without ever obtaining a success.
Examples would be
1) flip a coin UNTIL you get a head
2) roll a die UNTIL you get a 3
3) attempt a three-point shot in basketball UNTIL you make a basket
a random variable X is geometric provided that the following conditions are met: (a-c are same as binomial)
a) each observation falls into one of just two categories, called success or failure
b) probability of a success, p, is the same for each observation
c) observations are all independent
d) the variable of interest is the number of trials required to obtain the FIRST success.
Recognizing the existence of either a binomial or geometric distribution is essential to know how to proceed with your data analysis. Here is an example that should help explain how to VERIFY a geometric setting.
An experiment consists of rolling a single die. The event of interest is rolling a 3: this is called a success. The random variable is defined as X = number of trials UNTIL a 3 occurs. To VERIFY that this is a geometric setting, note that rolling a 3 will represent a success, and rolling any other number will represent a failure. The probability of rolling a 3 on each roll is the same: 1/6. The observations are independent. A trial consists of rolling the die once. We roll the die until a 3 appears. Since all of the requirements are satisfied, this experiment describes a geometric setting.
Rule for calculating geometric probabilities:
If X has a geometric distribution with probability p of success and (1-p) of failure on each observation, the possible values of X are 1, 2, 3, .... If n is any one of these values, the probability that the first success occurs onn the nth trial is
P(X=n) = (1-p)n-1 p
This rule can be used to construct a probability distribution table for X = number of rolls of a die until a 3 occurs from our earlier example. We'll use the TI 83 to do this now. When graphing the distribution of X as a probability distribution histogram it will appear to be strongly skewed to the right. This will ALWAYS be the case. Try to determine why from the formula!
The mean (expected value) and standard deviation of a geometric random variable can be calculated using these formulas:
If X is a geometric random variable with probability of success p on each trial, then the mean of the random variable , that is the expected number of trials required to get the first success, is
m = 1/p and the variance of X is (1-p)/p2 whose square root yields the standard deviation
One more rule to go....
P(X >n) or the probability that it takes MORE than a certain number of trials to achieve the first success.
P(X=n) = (1-p)n
Since we are seeking the first success at whatever trial it occurs, geometric simulations are called "waiting time" simulations. Makes sense! Let's discuss the cereal problem on page 473.