Parameters of a distribution of a physical variable

[tworitdash]

Pardon me if this is a very silly question. Although my research involves a lot of probability distributions, I consider myself a fledgling statistician.

When people assign a probability distribution to a variable in a physical process, is it inherently assumed that the parameters of this distribution are not related to each other?

My intuition says yes. If for example I assign a variable with a two parameter distribution and they are somehow dependent on each other, then it should be formulated as an one parameter distribution instead. Am I correct?

I have a similar situation in my research and I found in literature that the diameter of raindrops is gamma distributed with two parameters $\eta$ (shape parameter) and $\Lambda$ (inverse scale parameter - inversely proportional to the mean diameter in a volume). In old literature people really considered them to be independent parameters, but the current literature in this field show quite different results. When they try to fit it with gamma they really see a dependency of these two parameters.

This dependency can not be explained by any sort physical theory. Then, should the diameter be modeled by a different distribution instead?

 

[FactChecker]

When people assign a probability distribution to a variable in a physical process, is it inherently assumed that the parameters of this distribution are not related to each other?

No. That completely depends on the subject. A subject expert may, or may not, say that there is a relationship between the parameters.

My intuition says yes. If for example I assign a variable with a two parameter distribution and they are somehow dependent on each other, then it should be formulated as an one parameter distribution instead. Am I correct?

If you know that for sure and there are no exceptions, then that sounds appropriate. The disadvantage is that any confidence intervals on the parameter would not be easily calculated from the standard, two-parameter distribution. But there is no easy way to avoid that problem if the parameters really are related.

I have a similar situation in my research and I found in literature that the diameter of raindrops is gamma distributed with two parameters $\eta$ (shape parameter) and $\Lambda$ (inverse scale parameter - inversely proportional to the mean diameter in a volume). In old literature people really considered them to be independent parameters, but the current literature in this field show quite different results. When they try to fit it with gamma they really see a dependency of these two parameters.

This dependency can not be explained by any sort physical theory. Then, should the diameter be modeled by a different distribution instead?

It is a different distribution. The theory of gamma distributions depends on the parameters being constants.

 

[tworitdash]

No. That completely depends on the subject. A subject expert may, or may not, say that there is a relationship between the parameters.

If you know that for sure and there are no exceptions, then that sounds appropriate. The disadvantage is that any confidence intervals on the parameter would not be easily calculated from the standard, two-parameter distribution. But there is no easy way to avoid that problem if the parameters really are related.

It is a different distribution. The theory of gamma distributions depends on the parameters being constants.

Thanks for the answer; I really appreciate it. What do you mean by constants here? For example, the claim is that the drop diameters are gamma distributed , but depending on different kinds of physical conditions the parameters of this distribution can vary (Like a different mean diameter in the volume or a different shape). However, for one specific physical condition, they are constants. The relation that people find between these parameters is when they arrange the data with all possible physical conditions. They see that there is either a linear and a quadratic relation between shape and the reciprocal of the mean diameter. I just explained this so that we are on the same page. If by constants you meant one type of physical condition, then we are on the same page and I think these parameters shouldn't be related to each other with varying physical conditions.

 

[FactChecker]

It sounds like I misinterpreted your original question.

What do you mean by constants here?

I mean that for each case where you do statistical analysis, each parameter has a single value.

For example, the claim is that the drop diameters are gamma distributed , but depending on different kinds of physical conditions the parameters of this distribution can vary (Like a different mean diameter in the volume or a different shape).

That is not a problem, as long as you just deal with one physical condition at a time for each analysis.

However, for one specific physical condition, they are constants.

Good. So for a given physical condition, you can do the usual analysis using the Gamma function and those particular parameter values.

The relation that people find between these parameters is when they arrange the data with all possible physical conditions. They see that there is either a linear and a quadratic relation between shape and the reciprocal of the mean diameter. I just explained this so that we are on the same page. If by constants you meant one type of physical condition, then we are on the same page

Yes.

and I think these parameters shouldn't be related to each other with varying physical conditions.

That is a different problem from what I was originally thinking. I think that you could apply the Chi-squared Goodness of Fit test to a large sample from a variety of physical conditions to see if you can reject their theory of a particular relationship between the parameters.
If you have data from a variety of physical conditions, you can use their theorized parameter relationship to estimate the number of datapoints that should (on average) have certain values. Then you can test whether the data fits that theory. It might require a lot of data to get a strong test.
ADDED UPDATE: Their theoretical model should not rely on the data you use for your Goodness of Fit test, otherwise their model might be rigged to pass the test for that specific set of data.

 

[tworitdash]

It sounds like I misinterpreted your original question.

I mean that for each case where you do statistical analysis, each parameter has a single value.

That is not a problem, as long as you just deal with one physical condition at a time for each analysis.

Good. So for a given physical condition, you can do the usual analysis using the Gamma function and those particular parameter values.

Yes.

That is a different problem from what I was originally thinking. I think that you could apply the Chi-squared Goodness of Fit test to a large sample from a variety of physical conditions to see if you can reject their theory of a particular relationship between the parameters.
If you have data from a variety of physical conditions, you can use their theorized parameter relationship to estimate the number of datapoints that should (on average) have certain values. Then you can test whether the data fits that theory. It might require a lot of data to get a strong test.
ADDED UPDATE: Their theoretical model should not rely on the data you use for your Goodness of Fit test, otherwise their model might be rigged to pass the test for that specific set of data.

Thanks for the reply again. I like the discussion. I also discussed with one of my professors, and I think for me it is very difficult to disprove the theories people made over the last 50 years. I see different relations between these two parameters, and all of them are empirical. And, I also don't have access to a sensor that can record all sort of conditions. I already have estimation procedures of my own to find parameters given the measurements from a sensor collected from different dates. I will keep these parameters independent and estimate them. At the end I will see a relation if there is any. Secondly, the estimation I perform has a likelihood function. It is a function of space. I can really assess these models based on a goodness of fit (based on my likelihood). This information will really assess if at all a Gamma distribution is adequate or not.