# Appropriate distribution for the given problem

Red blood cell deficiency may be determined by examining a specimen of the blood under a microscope. Suppose that certain small fixed volume contains on an average $20$ red cells for normal persons. What is the probability that a specimen from a normal person will contain less that $15$ red cells ?

I thought of using the Exponential distribution for this problem since a single parameter is involved and the average number of cell is given.

Thus , $N$ denotes the number of red cells present in the specimen of a normal person. So , $N$ ~ $Exp(\lambda)$ , where $\frac{1}{\lambda} = 20$.

Hence , $P(N < 15)$ can be easily calculated. But as far as I know exponential distribution is used whenever some 'time' related or a 'process' related thing is going on , here although I was able to get a solution , but that doesn't seem correct. Does it ?

• Note that you're dealing with a count rather than a continuous process. Oct 10, 2016 at 19:12
• So any suggestions for what distribution should be used ? I know Poisson can't be applied here. @Glen_b Oct 11, 2016 at 10:22
• If you give the reasons why you know that the Poisson cannot be applied, it might help identify an alternative. Oct 11, 2016 at 10:48
• That's a situation in which the Poisson is natural certainly, but it's a perfectly reasonable model in a variety of other situations. In this case, you have something quite analogous to the situation you describe. "Fixed interval of time" becomes "fixed volume", and my guess is you could reasonably treat the inclusion of the individual cells as independent events. But the Poisson may be a reasonable model even when you can't draw such an analogy. ... ctd Oct 11, 2016 at 22:25
• ctd ... I don't know for sure that it's reasonable here (I have no experience modelling this situation having never seen real data for it -- maybe there's some characteristic of what is going on that means it's better modelled as geometric or something), but my first thought would have been Poisson. Oct 11, 2016 at 22:25

Suppose that certain small fixed volume contains on an average 20 red cells ... I thought of using the Exponential distribution for this problem since a single parameter is involved and the average number of cell is given.

IMHO, your second statement doesn't quite follow from the first. It's unclear that a single parameter (specifically the mean) is involved. A single parameter (specifically the mean) is what's been recorded and given here. Given samples generated by any distribution (including those with multiple parameters), you could just state what the average happened to be.

There are two non-exclusive ways you could go.

The first is by trying to deduce the distribution through the underlying process. For example, looking at erythropoiesis, one could start with the following model (I'm not a biologist, so the details might be wrong - this is just an example). Stem cells generate blood cells at an extremely-high fixed rate, each cell is ready in about 7 days, and lives for for a number of days that is normally distributed. Under the assumption that the blood volume is approximately constant, the density of blood cells could be approximated as normal.

For other underlying processes, I'm guessing the distribution could be that of a queuing system or a Poisson distribution.

The second is by starting with a specific distribution (or small number of distributions), and testing the hypothesis that the sample is generated by the distribution. See here how to check if a process is normal, see here how to check if the process is Poisson, and see here how to check if a process is exponential or normal.

Once you decide which distribution best fits the data, you will probably need to estimate the parameter(s). See here, for example, how to estimate the parameters of a normal distribution.

• But isn't the poisson process applied where a certain event is happening periodically ? Like say , consider a telephone booth , which receives , say , $40$ calls per hour. Here a poisson process is suitable. But I still can't figure out how you concluded it as a poisson distribution. Oct 11, 2016 at 16:49
• @User9523 A Poisson process is applicable when something like phone calls come in with an exponential process describing the time between calls. I can imagine stuff like hypothetical thin blood vessels where red blood cells appear with exponential waiting times in between, but I don't have the bio background to know if this happens / is relevant to blood samples. Oct 11, 2016 at 18:24
• I gave it a try in the answer section below ! Oct 12, 2016 at 9:58

Let us say ,on an average 20 red cells are present per $x$ unit volume.

Thus , if $N$ represents the number of red cells we can have : $N$ ~ $Pois(\lambda)$, where

$\lambda = 20$ cells per $x$ unit volume.

Hence $P(N<15)$ can be calculated.