How to find a probability distribution the parameters of which do not impact each other like mean and variance in normal distribution do?

I need to find a probability distribution to fit my data. My data has two important features, duration and activity count. Duration means how long one sequence lasts and activity count means the number of activities in one sequence.

I want to draw a curve, which should be (but not definitely necessary) like normal distribution. The height of the peak is related to the activity count. The breadth of the peak (confidence area) is related to the duration. In my context, more activity count means higher peak, but the breadth has no change. Also, longer duration means wider breadth, but the peak has no change.

In other words. peak is just related to the number of activities, while the breadth is just related to the duration.

So, you can tell the difference between normal distribution and the distribution I expected. For a normal distribution, variance will change both the breadth of the peak and the height of the peak, which is not what I want.

For example, on the above figure, the same height means that the number of activities are the same for two sequence. But one sequence has longer duration than the other, so the blue curve's peak is wider.

If one distribution cannot do this, can i use high dimensional distributions or the combine of multiple distributions to do it?

So, could anyone suggest a probability distribution for me?