We are building EduLadder(ELADR) - Protocol

The Eladr Protocol is a decentralized, security and efficiency enhanced Web3 noSQL database powered by IPFS as the data storage layer https://ipfs.io/, and the Cardano block chain as the rewards token platform, https://cardano.org/. It provides a JSON based, IPFS layer 2 solution for data indexing and retrieval in an 'append only' file system built with open source Node.js API libraries.

The ELADR token was designed to incentivize and reward community members as a proof of contribution. Token holders are also granted access to EduLadder.com premium features as well as associated ELADR token enabled apps.


Real Problems! Real Experts!

Join Our Telegram Channel !

The Eduladder is a community of students, teachers, and programmers. We help you to solve your academic and programming questions fast.
In eduladder you can Ask,Answer,Listen,Earn and Download Questions and Question papers.
Watch related videos of your favorite subject.
Connect with students from different parts of the world.
Apply or Post Jobs, Courses ,Internships and Volunteering opportunity. For FREE
See Our team
Wondering how we keep quality?
Got unsolved questions? Ask Questions

Engineering-Physics-06PHY12--BE-I-Semester--VTU-Belagaum-Unit-1-Modern-physics-->View question

Asked On2017-06-12 05:04:26 by:milan-ransingh

Taged users:

Be first to like this question

Be first to dislike this question
Talk about this  Like  Dislike
View all questions

Probability Density

Probability density is a concept that naturally arises whenever you talk about probability in connection with a continuous variable, such as position of a particle. In contrast to discrete probability (such as idealized coin-flipping or dice-rolling), we can't directly assign a probability to each individual outcome. Probabilities have to sum to one, but there are infinitely many possible positions for a particle, so any individual position would have to have probability zero. So how can we talk about probability spread over a continuous domain?

Well, we can talk about the probability of finding the particle within some region. If the region covers a nonzero length/area/volume/whatever, we can expect it to have a nonzero probability. But the exact probability will generally depend on the size/shape/position of the region in a complicated way.

Suppose we consider a very small region. To be concrete let's suppose we're talking about a 1D position and our small region is some interval [x,x+dx]. The interval has width dx. It stands to reason that a small interval will have a small probability. Moreover, if probability is "spread smoothly" in a sense that I won't bother making precise (hey, this is Physics.SE, not Math.SE!), then for small intervals the probability will be proportional to the width of the interval. If you halve the size of the interval you get half as much probability. Therefore

where dP is the small probability of finding the particle in this small region, and p(x) is some proportionality constant. I've written it as a function of x because the proportionality could vary from point to point.

Well, p(x) is called the probability density, and it has the units of probability per unit length. (Or unit area, or volume, depending on how many dimensions we do this in.) It's a lot like mass density, which is mass per unit length/area/volume, or charge density is charge per unit length/area/volume, etc...

So, now you can imagine calculating the probability to find the particle in any arbitrary region, by slicing up that region into many small ones and summing up dP, which is p(x)dx, for all of them. Well, this is just integrating:

P(particle is in region R)=Rp(x)dx
In particular, the particle has to be somewhere, so p(x) must integrate to one: