I am trying to understand kernel density esitmation (week 2 - Visualizing data: Kernel density estimation).

I feel I have understod the idea quite well except for a few details, and I would really appreciate if someone who understands more clearly than me could explain.

We have n original data points, we draw a guassian distribution at these points, then we sum the guassians to get new points that reflect the overlap of data.

Q1. Do we sum only at the n original data points, or (as other sources suggest) at fixed intervals across the x-axis?

Having done this, the next step in the lecture is to ‘multiply everything by 1/n and sum the curves’.

Q2a). I presume this division by n refers to the n original data points, and so suggests that the summation question in Q1 is indeed only done at the n original data points, but I don’t see why dividing by n at these n data points guarantees a continuous PDF who’s area sums to 1?

Q2b) If the summation of guassians doesn’t only occur at the n original data points and is actually done at, say, m regular intervals, then would the multiplication be by 1/m?

Q3. Even if it is done at m regular intervals and everything is divided by m, I don’t quite understand why this would result in a PDF with an area of 1, when it seems that the area is determined less by the height of the n points, and more by the paramaters of the original guassian kernal. Or is it simply a given that this occurs?

Apologies if this is a basic misunderstanding.

Many thanks for any help anyone can give