Lets say that I have measured temperature at three different weather stations and define them as three random variables X,Y,Z. One is in a valley, one on a mountain and one in a city. They are not independent since they are located close to each other. However their mean and StDev differs between the stations.

Based on these measurements I estimate their density functions.

Now I'm interested in finding the sum of all the density functions.

If they were independent that would be the convolution of the separate density functions. But there not. My books only discuss the case where the variables are independent so I dont have a clue how to solve it.

Can anyone help please?