If your speed is 0, you'll take all the rain until it stops.

If you run at infinite speed, you will take all the rain that is in the area you are going to pass through.

If your speed is in between 0 and infinite speed, you will take all the rain in the area you are going to pass through plus the rains that falls and hits you.

The faster you go, the lesser falling rain you are going to receive.

Thus it is better to run. Is it mathematical enough?

About experiments: it doesn't make sense to use cotton clothes or similar. Take 2 equals recipients, and keep them well exposed to rain.

Move one at walk speed, move the other at run speed, for the SAME amount of space. Then measure. I guess the running one will have lesser water.

On the other hand, if you fix the length of time of rain exposure, running will mean more water, because you will pass through more space with rain inside.