It is widely accepted that bacterial spores germinate when the germinant(s) bind to receptors and activate the hydrolysis of the spore coat that leads to germination. There are only an average of nine binding sites per spore distributed among the population according to the Weibull function. The number of receptor sites is clearly discrete since there can be no fractions of a receptor and you can divide the spore population into subpopulations each of which has the same exact number of receptors.
The Weibull function is, however described (in Wikipedia) as a continuous function.
The average number of receptors is fractional and the average number of receptors within a sub-population that have a germinant molecule bound to them is also fractional so both these are continuous variables.
I've developed a mathematical model based on this conceptual model that describes the kinetics of germination. In developing the model I used differential equations which are later integrated.
When I was taught statistics it seemed that distributions were either discrete (eg Poisson) or continuous (eg normal) but never both. Is this still true or is it now regarded as hopelessly old fashioned and pedantic?
The development of the kinetic model starts by considering a generalised sub-population of spores and then using the distribution function to expand it to the whole population. So I start with the Weibull function as a discrete distribution and it metamorphoses into a continuous distribution in the final equations.
Is this acceptable or, if not, what is?
In general, discrete distributions corresponding to continuous one have been developed, for example geometric distribution is discrete analogue of continuous exponential distribution, binomial distribution is discrete analogue of gamma continuous distribution.
The difference is that discrete distribution is characterized by probability mass function and continuous one from probability density function, mathematically. From practical point of view they describe similar phenomena, which for some conditions appear discrete in nature and some others continuous.
In your case, there exist discrete Weibull distribution analogue to continuous Weibull distribution. I am sending you the link to the paper.
http://www.hindawi.com/journals/jps/2013/946562/
The answer you have already received is fine and should be helpful in solving the problem you posed, but I would like to comment about this statement that you made: "Is this still true or is it now regarded as hopelessly old fashioned and pedantic?" There is nothing pedantic or old-fashioned about appropriately distinguishing discrete random variables from continuous ones: No passage of time, or modernity will change the fact that if a random variable can take on only integer values (number of defects per unit area, for example), it is discrete; and if a random variable takes values on the real line, (waiting time until the occurrence of an event, for example) it is continuous.
I believe that the source of your "dilemma" may be because while you are dealing with discrete variables that take on integer values, you are also concerned about averages (at least that is what I can gather from your brief description here, which, admittedly, had to be compact). Distribution of averages of random variables, regardless of the nature of the original random variables themselves, are always continuous, for the obvious reason that averages of integers are continuous. (Under some circumstances, it is actually possible to show, via the Central Limit Theorem, that the distributions of the average of a random variable tends to be Gaussian.) The key therefore is to focus on the fundamental nature of the random variable in question: a discrete random variable will always be described (appropriately) by a discrete probability distribution function, and a continuous random variable by a probability density function--facts that will never be any more "old-fashioned" than it will be for an integer somehow to become classified as a real-valued variable.
Babatunde Ogunnaike – Thank you for this. I am now confident that my model is statistically proper, appropriate and valid and that I can describe it clearly without solecism. I am not so sure, however, that I can justify the use of the Weibull function, other than by showing that the distribution of receptor sites must be asymmetric. The Weibull’s virtue, that it can be made to mimic almost any distribution system by setting suitable values of the shape and size parameters, can also be a problem because it can be adjusted to allow a model in which it is employed to fit almost any kinetic data. I think I can use the model with experimental data to determine the values of parameters under different conditions so that I can check for internal consistency.
Not knowing as much about the true phenomena under-girding the distribution of receptor sites, may I suggest that you may find the gamma distribution to be more appropriate? Here is my reasoning: In Chapter 9 of my book (Random Phenomena: CRC Press, 2009) I show how the gamma density function can be used to model the distribution of distances between DNA replication origins in cells. Figure 9.3 in that book (page 270) shows a gamma density fit to data obtained from the budding yeast S. cerevesiae genome. To the extent that you may be able to argue for similarities between your problem and this one, then the gamma distribution might provide a justifiably fit.
I have looked at the gamma function, which has the same sort of versatility as the Weibull but a slightly different shape when the skewness is to the higher values. It will be interesting to compare them using my data.
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