Probability vs Density

Part Three: The PDF (Probability Density Function)URL copied

Here's the thing about continous distributions - there are infinite values in any bucket. Hence, a PMF with infinite resolution would give a decent approximation of the shape of the function even if it is at the cost of near-zero heights. So one could compromize on a decently large resolution and freeze the PMF so as to get the shape. Then the y-axis is rescaled such that the area under the shape becomes 1. In our case, this results in a rectangle with width 0.5 and height 2. This rescaling of y-axis itself is the normalizing factor, the per-ness in the density #bullshitspiration.

On a side note - Kernel Density Estimation - is all about finding the most accurate PDF from given observations/histogram/PMF. I used an r-based function called 'density' to arrive at the above graph which uses a very simple technique

Now how does one make sense of the graph? What does the 2 even mean?

Let's do a comparision with another simple number generator, this time the numbers are equally probable between 0 and 1. It's PDF would look like as follows. The curve is much flatter, obviously, because the width of the rectangle is larger (remember the area has to be 1).
What does this say? The likelihood of generating a number around 0.2 (or any number between 0 and 0.5 because the PDFs are flat) is twice as much in our first distribution as compared to the second.

Let's end with another example.

Given above are two probability distribution functions. We may know nothing about their statistics, but somethings we could say by simply looking -

1. The probability of getting/generating/finding a number around 0.0 in distribution 'x' is twice as much as getting a number around 0.0 in distribution 'y' (We are simply comparing the heights).

2. One is 4 times as likely to get a number around 0.0 than getting a number around 0.2 in the distribution 'x' (Now we are comparing outcomes from the same distribution. At 0.0 the height is ~4 and at at 0.2 it is ~1)

3. One is equally likely to get a number around 0.2 in 'x' and 0.3 in 'y' (The heights are pretty much the same)

So, there. Now you (I?) know that density by itself has no meaning. Now you know the philosophical meaning when Lao-Tzu says

So it is that existence and non-existence give birth the one to (the idea of) the other; that difficulty and ease produce the one (the idea of) the other; that length and shortness fashion out the one the figure of the other; that (the ideas of) height and lowness arise from the contrast of the one with the other; that the musical notes and tones become harmonious through the relation of one with another; and that being before and behind give the idea of one following another.