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Archive for the ‘Statistics’ Category

I don’t know how many of the readers here pay attention to what’s happening in Wall Street. Yesterdays trade was quite spectacular, although from many points of view it is terrifying and it is a day that will probably live in infamy.  From the academic point of view I’m sure it will be studied to exhaustion.

All in a day's work.

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Sometimes you work hard on a project, and you have some theoretical framework that explains how things should behave in certain limits and how there should be a natural expansion to be able to do a fit for the limit one is studying. Then sometimes you take some data, let us say you do a simulation, and the data just does not conform to these expectations.

You then get stuck with data that you don’t really know how to analyze. And it is terribly frustrating.

You can try to understand what direction the data is pointing to, but it can be more of a ‘nebulous oracle’ than a clear straight arrow pointing to the path you should take. This is normal: it happens all the time.

Think about that.

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As I told you previously, my project for the summer was to learn some Python language. I’ve been toying with it for a bit and with a few lines of code I was happily getting output. It’s great that I don’t have to declare types before using variables! And it took me a while before I realized that indentation substitutes for all the curly brackets. So far, I find it rather intuitive.

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Probability game

Here is a fun game of probability and statistics that I was told about this week. You have a game where you toss a coin and you win if you get three heads in a row. What is the average number of throws for a win?

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Over at Cosmic Variance, they had a recent post on big surprises that one could have in physics. In the comments someone suggested that we should be looking for messages from the creator in the digits of pi. I’m sure this was said in jest, but I’ve seen enough similar attempts forwarded into my e-mail box to know that this goes on.  Of course, this is just one more version of what we in physics pejoratively describe as numerology: some random collection of facts about some really important mystical number, that has no physical mechanism to describe a physical situation . Apart from pi, another very popular set of such numbers are 42 and 137. The plan of this approach is that everything there is to know about the world is encoded in these numbers, if you only have the correct algorithm to decode it. One of the popular algorithms is to look for the digits of pi or some other irrational number and to try to see patterns in them.

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Probabilities in physics.

Many people get spooked by the fact that in physics, especially in quantum mechanics, the only predictions that we can make are probabilistic in nature. Let us say you have a polarized photon, and you ask if it will pass through by another polarizer or not. The only answer you get is it will pass 60 percent of the time and that it will not pass 40 percent of the time. Moreover you can not tell in advance in which way it will go.

 

This is a fundamental fact about the world we live in. That fact does not mean that physics is unpredictive. Quite the contrary, you can predict those probabilities with remarkable precision. 

However, this also means that making precise measurements in physics can require a lot of statistical manipulation, because changing the underlying physical parameters can only change the probability distributions for events, and even though some events might become more unlikely, they are not impossible. It might be instead that you got a freak observation (unlikely events do happen all the time).

So I though I would give you a taste for how to make predictions by giving you a statistical puzzle. 

Let us say that someone gives you a lopsided bet. Such that with probability r<1 one gets heads, and with probability (1-r) one gets tails, and you have to pick heads or tails. 

You only know the outcome of the first event. After the first toss it came out heads let us say.

Now for the question I ask of you. What is the probability that r>1/2 ?

 

Excuse me?-You might ask- I thought the parameter r was a given.

 

And my answer is that, yes, someone else knows r, but he won’t reveal the answer, because if he does so he could lose a lot of money, so you have to make assumptions about how r is distributed to be able to answer this question in a meaningful way. Seeing as you have no information about r, you might as well assume that any value of r is equally likely and then the question above makes perfect sense.  Go have fun on this one.

 

Incidentally, similar problems do show up in physics all of the time. Let us say that the parameter r gets replaced by an angle of a polarizer (as above) and that your job is to find the best estimate for that angle with the knowledge of only one (or 5 or 15) polarized photons… If you ask astrophysics folks about polarization of light of very distant stars every single photon counts.

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OK, so I admit I occasionally wander around and fill random tests on the web as well. It seems a lot of people do, so what is the harm in admitting it?

This is not a rhetorical question. One might do a bit of social calculus: does filling this test say something about me? What will my colleagues think of me for admitting it? Will I go up or down on their esteem? By how many points? Will they forget quickly or will they remember this and tease me forever?

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