# Making probability density function of scatter with normal distribution function

I want to make probability density function of gas scattering.

The precondition of the model is,

1. With out collision, atoms fly straight.
2. We have many atoms, so it can be treated statistically.
3. Atom flies from source to target, and the distance is 150mm, (let d = 150mm)
4. During the flight atom experiences collision every 25mm, which changes atom's direction left or right with the probability of half and half. (let $$\lambda$$ = 25mm)
5. Collisions are independent each other, that is, collision don't influence each other.
6. It's totally elastic collision. So, only mathematical approach is needed here

I want to find probability density function on the target as a function of a-xis (so called x) that cross the target(normal to straight line from source to target),

$$f = f(x)$$

Finding this function, I though, normal distribution is suitable (because, it is a probability density function describing independent trial)

I started from normal distribution

$$f(x) = \frac{exp[-m^2/n]}{\sqrt{\pi n^2}}$$

when, S: standard deviation m: m th position n: number of collisions (and it is $$n = 150/25 = 6$$)

And I induced m and n by this logic,

$$m = x/\lambda$$

$$n = d/\lambda$$

substituting these into, eq. above..

$$f(x) = \frac{exp[-x^2/d\lambda]}{\sqrt{\pi (d/\lambda)^2}}$$

I found that variance of this function decreases when $$\lambda$$ is decreases. But I think, variance of the function should increase when $$\lambda$$ is decreases, Because the number of collision, n increase as $$\lambda$$ is decreases and it make an atom scatter further.

What's wrong with it?