# Modelling values by overlaying existing distributions

Imagine I have two datasets, one has values of a dependent variable, time spend walking, along with many other independent variables such as for instance gender, age group, day of week and dog owner etc. The other has a lot less data, just the independent variables and the number of people within these groups who were recorded as walking for a particular combination of gender, age group etc.

My question:

If I use the first dataset to construct various distribution of walking times for each level of the data e.g. one distribution for males, one for females, one for Wednesdays etc. (provided sufficient data points – I’ve read 100 is a good rule of thumb)

Would I then be able to use these distributions to obtain distributions and subsequently (via something like inverse transform sampling) a collection of data points for each row of the second dataset. e.g. could I get a distribution of walking times for females, aged 40-50, on a Wednesday who owned a dog by overlaying the underlying distributions?

Edit:

I have not seen the data yet but I am almost certain the distribution of ‘walking times’ would be bimodal, with a large peak at the beginning and a smoother peak at the end with some overlap in the middle. This could possibly be modeled by an overlay of two skewed Gaussian distributions.

These underlying distributions can not be attributed to an feature I have access to (or that exists). i.e. if I were to draw a distributions at its most granular level say the distribution of walking times for males, age group 18-25, Wednesday and w/o a dog. These two peaks would still appear. However, I expect the relative size of the two peaks, and separation between them to be influenced by the features. E.g. dog owners would have a larger right hand peak than non-dog owners.

This is what I hope to model by overlaying various distributions at the top level to obtain distributions at the most granular level.

Many thanks, J

• The short answer is yes--but arguably the question is too broad to be accommodated in our restricted format. The underlying model is called regression and what you are doing is called fitting the regression model to the data. The tricky part concerns making principled statements about how uncertain those fitted distributions are, how to quantify that uncertainty, and interpreting the results accordingly. There are a huge variety of possible regression models and associated fitting procedures. – whuber May 15 '19 at 17:51
• Thanks again @Whuber, if my independent variables are all categorical is there any specific models you would recommend. I've been reading about mixture models and think it might suit my purpose, what do you think? – JDraper May 16 '19 at 13:36
• You need to explore the data first: you haven't presented enough information to enable anyone to recommend a model and justify that recommendation. – whuber May 16 '19 at 13:59
• Sorry. I haven’t been very clear. Hopefully the edit helps. – JDraper May 16 '19 at 15:17