Determining whether a successive instances are conditional or independent I hope I can appropriately describe the problem.
I have an example data set with 610 individuals with made up analogy (it's not really about cars).
I want to determine the number of people have x numbers of cars, and get the following data set.
# cars    # people
0         491
1         101
2         14
3         3
4         1

When I graphed the data with x as the # of cars and y as the # of people, I get one particular graph. What caught my interest was when I scaled the y-axis on a log scale. Once I did so, I found a relatively linear relationship. This brings me to what I want to test. If having a previous car is conditional upon having a previous car (and I am presuming it would depress the likelihood), I should not see a linear relationship. However, if the chance to have a second car or more is based on successive chances of a car being independent, I should see a linear relationship at log scale.
How would I go about testing this? Either by testing the hypothesis that the successive car probabilities are independent, i.e. the chance of getting car 2 is the same as getting car 1. Or by testing the hypothesis (and failing it) that the successive car probabilities are dependent on previous criteria.
 A: The geometric distribution has the memoryless property you think you observe and would be consistent with a linear log frequency. To test this hypothesis, first make sure it would be consistent with theory about the phenomena, e.g. do we have an economic theory of car ownership behavior that would support that hypothesis, then use a $\chi^2$ to test to make sure your observed data does not violate it.
We use one degree of freedom to estimate the geometric distribution parameter $p$ in the $(1-p)^k p, k=0,1\dots$ parametrization of the pmf. The sample mean is $0.2328$ cars, so we'll estimate the $p$ parameter to be $p = 1/(1+0.2328) = 0.8112$. Now the actual and expected frequencies are

k           0           1           2           3           4       >4
p(X=k)      0.8112      0.1532      0.0289      0.0055      0.0010  0.0002
exp.        494.8138    93.4356     17.6434     3.3316      0.6291  0.1464
act.        491         101         14          3           1       0
sq. dev.    14.5453     57.2203     13.2745     0.1100      0.1376  0.0214
sq dev/exp  0.0294      0.6124      0.7524      0.0330      0.2187  0.1464

And the sum of the ratio of squared deviations to the expected is 1.7923, our $\chi^2$ statistic with 4 degrees of freedom. If we go by a 0.05 level of significance, then the critical value is 9.4877. So, we do not have reason based on this data at our preselected level of significance to reject our hypothesis that this data corresponds to a geometric distribution.  
Note that this is insufficient to establish a theory of car ownership. We have only shown that this data is not inconsistent with our hypothesis.
