# How to decompose a distribution with two peaks?

I'm modeling saving account's amount, whose change looks like a log-normal distribution. It means suppose $y$ is total saving account's amount; $x = \ln(y)$ is the natural log; $dx$, the daily change, seems follows normal distribution.

However, $dx$ seems has 2 peaks, one is high and a bit less than 0, one is lower and bigger than 0.

The reason could be that people are likely to frequently withdraw little amount of money, e.g. weekly withdraw 200 bucks for everyday expense, and deposit more money less frequently, e.g. monthly deposit the paycheck.

Now the question is, how could this distribution be decomposed?

Seems I can't decompose it to two normal distribution, as two independent normal distribution's sum is still a normal distribution.

Any suggestion?

• "seems i can't decomposite it to 2 normal distribution, as 2 independent normal distribution's sum is still a normal distribution" -- you're looking at something more like a mixture, not a convolution. Each change is not a sum of a deposit and a withdrawal, it's either one or the other. – Glen_b Jul 25 '13 at 6:04

## 1 Answer

As far as I understand your problem, I believe that you should use a mixture of two normal distribution. You assume that some of your observation behave according to the first model and some according to the second model and you estimate the parameters of each model and the probability to behave according to each model.

One good example of an application would be the case where you have data about height for a sample of males and females but unfortunately your variable for the gender of the person has been deleted. In that case, you would have a mixture of two normal distributions.

See Wikipedia to learn more about mixture models