## probability – Density function of flip the coin twice

We have a rigged coin, the probability of obtaining heads is triple that of obtaining tails.

Consider the variable $$X$$ defined as follows: We flip the coin twice in a row.

• If Heads are obtained on the first roll and Tails on the second, we take $$X = 1$$.
• If you get Tails on the first roll and Heads on the second, we take $$X = 2$$.
• Otherwise, the coin is tossed twice.

I want to find the density function of the variable but I’m confusing with that “Otherwise” means, the probability of getting head first and then tail is $$frac{3}{4}*frac{1}{4} = frac{3}{16}$$, same for the first tail and then head.
So $$P[X=1] = 3/16$$ and $$P[X=2] = 3/16$$, is this enough or I’m missing something? Any help is appreciated!

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## marching cubes – How to construct a octree density from SDF for MC

It very clear how it works with a regular grid. 3 inner loops – x, y, z over some size. The smaller the cells, so will be the mesh more detailed.

But how about octree. I know i can stich different levels with transvoxel (not sure with what max difference), but no idea how to construct the tree from SDF. It kind of needs to know if a cell is occupied and then split and so on? But how?

Or maybe octrees are bad idea for marching cubes and isn’t practiced (then what about ray cast)?

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## What is the difference between pixel pitch and pixel density?

It’s just a reciprocal relationship. Where pixel density measures, say, the number of pixels per inch, pixel pitch measures the number of inches per pixel (or the center-to-center spacing between pixels).

Pixel pitch expressed as a linear measurement, and if the pixels are non-square you may see two values specified. Pixel density, on the other hand, may be expressed as a linear measurement (pixels per inch or millimeter or what have you) or as an area measurement (pixels per square inch or pixels per square millimeter) — your equation assumes an area measure of pixel density and square pixels, and converts to a linear value.

I should probably add that pixel pitch is usually used to state device specifications (screens or sensors) — you wouldn’t often see it used to describe, say, the resolution settings you used to print an image.

## gaussian – Measuring probability density function for a device with two stochastic sources

We have a time delay element that given an input pulse it generates an output pulse after time T_nom. T-nom is the average time it takes for the device to output a pulse. Due to the device’s mismatching characteristics the delay has stochastic nature and follows a Gaussian distribution as following:

$$F(t)=frac{1}{sqrt{2pi} sigma_{mis}}e^{-frac{(t-T_{nom})^2}{2sigma_{mis}^2}}$$

where $$sigma_{mis}$$ is the device’s mismatch standard deviation. In addition to the device’s mismatching we have also what’s called a random noise or jitter which is inevitable for the device. This jitter, which additionally perturbs the delay time, also follows a Gaussian distribution function with the standard deviation $$sigma_{jit}$$.

What I want now is to include the effect of the jitter in the function F(t) and come up with a compact model to fully describe the stochastic behavior of the device. But how?

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## usability – What information density is reasonable?

The most general answer of course is, “it depends.” Who are the people using the product? What are they trying to achieve? Are they using it on a desktop or mobile phone? Are they using a mouse, touch screen, or keyboard to navigate (or other assistive technologies)? Is there flexibility for them to increase or reduce the density to what suits them?

Here is an interesting article I found on UX Collective “How white space killed an enterprise app (and why data density matters)” written by Christie Lenneville and Patrick Deuley.

Everything that article says is a good point and says it better than I could say, so I recommend checking it out.

You also need to ensure accessibility for those with vision, mobility, and/or cognitive considerations. Make sure everything is keyboard accessible. Make sure tables are marked up semantically and read correctly by a screen-reader. Make sure color contrast is compliant. Make sure text and content reflows so nothing overlaps or gets cut off when users zoom in or resize their screen. Read up on WCAG 2.2 for specifications on making sure content is readable and accessible no matter how dense or sparse.

And do thorough user interviews/observations and usability testing. If this is a redesign, have the users perform tasks on the previous/current version and time it. Then have them perform the same task in the redesign/prototype and time it. Did it save time? Did it prevent costly errors? Did it improve retention or conversions?

I think the best framework is around the UX research, though I know that doesn’t provide the kind of specific answer you were looking for. The most reasonable density is the one that allows people to do complete their task(s) most efficiently and without error, frustration, or the need for home-made workarounds.

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## K&F Concept ND2-ND32 Neutral Density Filter for Canon EF 100mm

For that particular filter at Amazon, it appears K&F Concept probably just provided the exact same “52 mm” descriptive text, regardless of which filter diameter is chosen.

The confusion here isn’t a filter or lens thing; it’s merely an example of technical issues, misunderstanding at the supplier end, or laziness in providing the correct descriptive text, with regards to e-commerce.

If you order the 67mm filter, it will fit your lens. If you order a 67mm filter but are delivered 52mm, then you were sent the wrong one through no fault of your own, and you should return it to Amazon and complain about the seller’s product page.

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## statistics – Measure of distance between probability density functions

I’m working on a project which deals with optimization of some sorts.

During the calculations, some likelihoods for parameters are computed exactly, but this is not always possible (as there might not be enough data available). So some likelihoods have to be approximated.

I would like to quantify the goodness of these approximations, by using them also in cases where all needed data is available and comparing the obtained likelihods with exact ones.
However, I’m not entirely sure what measure would be apt to quantify the “distance” between these different PDFs (apart from the obvious comparison of moments ect.), as this is not really my field.

If anyone has good recommendations or can point me in the right direction it would be greatly appreciated.

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## plotting – Rotation density plot with fixed Frame

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## Transformation probability density – Mathematics Stack Exchange

We let $$X in (0,1)$$ be beta-distributed with β-parameter 10 so that the density function is $$p(x)=10x^9$$. We let $$Y=sqrt{X}$$. Then I have to find the probability density of y i.e $$q(y)$$

I have found this I think I can use
$$q(y)=p(t^{-1}(y))|frac{partial}{partial y}t^{-1}(y)|$$
I think I get that $$t^{-1}(y)=y^2$$ and therefore $$p(t^{-1}(y)=10(y^2)^9=10y^{18}$$
So the probability density will be:
$$q(y)=p(t^{-1}(y))|frac{partial}{partial y}t^{-1}(y)|=10y^{18} cdot 2y =20y^{19}$$
While since y is always positive we can remove $$|cdot |$$. Is that correct? I’m not totally sure. It seems easier than in my lectures. Can anyone help me?

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## pr.probability – Is the set of probability measures on $mathbb{R}$ absolutely continuous with bounded density a closed subset?

The answer is yes. Indeed, a probability measure $$mu$$ over $$mathbb R$$ has a density bounded by a real $$K>0$$ iff the cdf of $$mu$$ is $$K$$-Lipschitz, that is, Lipschitz with the Lipschitz constant $$K$$.

So, you have a sequence $$(mu_n)$$ of probability measures over $$mathbb R$$ with $$K$$-Lipschitz cdf’s $$F_n$$ converging to the cdf $$F$$ of a probability measure $$mu$$ at all points of continuity of $$F$$. Since the set of all points of continuity of $$F$$ is dense in $$mathbb R$$, we conclude that $$F$$ is $$K$$-Lipschitz. So, $$mu$$ has a density bounded by $$K$$.

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