In biology experiment we often collect a sample of animals and measure their lengths. The length of a animal in the population is a random variable, so that we can estimate the probability mass function (if we treat length as discrete intervals) by counting the proportion of animals fall into each length intervals.

In the meantime, the weight of each sampled animal is also measured, and we could establish a similar "probability distribution" like function by computing the proportion of animal weight fall into each length interval.

It is very confusing to me because very often this -proportion of weight at given length distribution- has a very similar shape as the length probability mass function. And we can easily estimate such distribution with parameters (e.g. assuming normal or lognormal).

My question:

Although it is not a probability distribution of the length variable, can I call it a probability distribution? Mathematically, the proportion of weight at given length is a mixture of length probability mass function and weight distribution at length, so in the end will it follow a distribution as well?

Can someone shed some light on this "distribution" and maybe give some suggestions about what I can do and can not do with such "distribution"? For instance estimating such distribution, comparing the distribution, model the proportion using predictors.


Perhaps what you're asking about is a conditional probability distribution. The way I would describe what I think you're trying to describe is the distribution of weight conditional on a given value of length. Conditional probabilities are often modeled with regression; in the usual case of ordinary least-squares linear regression, the variable being conditioned upon is in fact treated as non-random.


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