Clarification about Wave-particle duality Okay,so I am learning about the double slit experiment done with electrons. I saw this picture, which shows the interference pattern being built up slowly with increasing number of electrons: 

I just wanted to confirm whether I have the correct understanding. The fact that the first image has a random distribution, shows that each electron interferes with itself and strikes a point on on the screen which would be dictated by the probability function. 
The interference pattern is the result of the same interference of many electrons and is a statistical property of many electrons.
Also, does this mean the electron travels as a wave, but then it obviously must strike as a particle since it hits a well defined spot on the screen?
 A: 
The fact that the first image has a random distribution, shows that each electron interferes with itself and strikes a point on on the screen which would be dictated by the probability function. 

What a) tells us is that a single electron was fired at two slits and was deflected to a point at an angle from a straight projections from the slits. The same would happen if one threw a billiard ball at two slits with the analogous sizes to the diameter of the ball.
b) and c) tells us that the shooter kept mostly hitting edges.
It is d) that shows a clear interference pattern in a distribution that answers to the question "what is the probability if I throw  electrons against  a double slit of appropriate dimensions that it will hit (x,y) on the screen."
The conclusion is that  an electron does not behave like a billiard ball, i.e. classical mechanics,  it does not have the behavior of a classical billiard ball when scattered.
This behavior is described accurately by solutions of the quantum mechanical equation with the boundary problem "electron scattering off two slits". The square of these solutions, called  wave functions, give us the probability distribution .
The statement "each electron interferes with itself" is misleading/confusing as far as the behavior of matter in dimensions where quantum mechanics prevails ( i.e. commensurate with h_bar). "The wave function describing the electron has interference terms passing the boundary of the two slits" is more correct. It is not a mass wave, nor an energy wave.
A: 
The fact that the first image has a random distribution, shows that each electron interferes with itself and strikes a point on on the screen which would be dictated by the probability function. 

Yes.

The interference pattern is the result of the same interference of many electrons and is a statistical property of many electrons.

Sort of. Each electron impact obeys (technically, samples) the probability distribution, which contains the interference. You need many hits for the probability distribution to become evident, but saying that the interference is exclusively a statistical phenomenon is slightly contentious.

Also, does this mean the electron travels as a wave, but then it obviously must strike as a particle since it hits a well defined spot on the screen?

Yes. There is a disparity in the evolution of quantum systems: wavelike, continuous,  and linear ("unitary") when they're left 'by themselves' and discrete, particle-like, discontinuous, nonlinear, when they're 'measured'. The current state of affairs is not really satisfactory, as there isn't an ironclad rule to say which situations are 'systems by themselves' and which situations are 'measurements', so there's still much to understand here. The overall problem is known as the measurement problem, and while there's been some impressive progress recently, we're still far from anything like a satisfactory understanding of these matters.
A: 
I just wanted to confirm whether I have the correct understanding. The fact that the first image has a random distribution, shows that each electron interferes with itself and strikes a point on on the screen which would be dictated by the probability function. 

More or less. It isn't quite random, and I'd say it's dictated by the nature of electrons which is modelled using a probability function. But yeah, sounds to me like you've got it. 

The interference pattern is the result of the same interference of many electrons and is a statistical property of many electrons.

As above. You could maybe chuck in the fact that the interference pattern is there as a result of each electron interfering with itself, and that you only see the pattern emerge when you send a lot of electrons through one after the other. 

Also, does this mean the electron travels as a wave, but then it obviously must strike as a particle since it hits a well defined spot on the screen?

Yep. That's the crux of it. You can see something similar in the optical Fourier transform, see Steven Lehar's web page: 

The electron wave goes through both slits, but when it's detected on the screen it gets converted into a dot. Then if you try to detect the electron at one of the slits, it gets converted into a dot and goes through that slit only, so the interference pattern disappears. You don't need any many-worlds multiverses to explain the dual slit experiment.            
A: From the theory of light waves we know that for similiar experiment an interference pattern occurs when light quanta interacts with the system. Now with electrons there is an unique "wave" for that particular experiment that guides those electrons that hit the screen. Initially electrons must have equal speed and direction for that clear pattern to emerge.
We can assosiate that "wave" $\psi$ with any electron in an abstract sense, and just like with light we have $|\psi|^2$ that stands for intensity of the hits on the screen (probability distribution). So for every electron there is a wave that depends on the system (e.g. double slit experiment, hydrogen atom, ...)
