So just a quick example of the transparency in r,

so here's a plot of a, a, of a thousand points.

And you could see that in the high density region

here, I'm using the pch 19 which is a solid circle.

In the

high density region, all the plot, all the points just kind of overlap each

other, so you just see a black,uh,blob there in the middle and it's,

card, kind of hard to see what the resolution for the different points

because there is no definition of points there because they are all overlapping.

So if I add a little transparency to the plot you'll see that the circles

become transparent so when they overlap each other

they, you can kind of see through them.

And so that the areas where

there are lots of points overlapping they, they get darker and when there are

areas where there are no points overlapping

you just see kind of a transparent circle.

So here I've got below, I've got the code that I've used to create this plot.

And you can see that I've specified an alpha parameter of 0.2.

And, and the r, the red, green and blue are all zeros so I can get black.

Uh,and so point two is pretty close to zero, so there's a lot of transparency.

Um,but you can see that when I use the transparency like thisiIt's

kind of a cheap and easy way to get essentially a histogram estimator

because, you know, the, the darker areas will represent higher density areas of

the plot and the lighter areas will represent kind of lower density areas.

So you can see a little bit more information when you have a plot with

a lot of different points in it when you use a little bit of transparency.