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HNRS 200 Probability in the Universe Lecture 1 Notes Throughout the class in addition to readings I will give you frequent handouts to supplement our discussion These will often contain additional informations and occasional references which are beyond the scope of in class discussion The Binomial Distribution In class we discussed the example of ipping a fair coin This means that each ip is independent of the previous ip and that each ip has a probability p 05 Now if all events are random then it is easy to see that if I ip a fair coin twice there are 4 ways that the results can come out HH HT TH TT And that only 1 of those combinations produces two heads Now consider Number of ways to get the outcome Probabilit of an outcome y Number of total outcomes In other words the probability of getting two heads in a row is 1425 The probability of getting one of each is 1250 and the probability of two tails in a row is 25 We call this sort of result a binomial statistic since in each case the coin can either come up heads or tails but it must come up one of them bi77 is the Greek pre x meaning two Now consider the probabilities if we ip many more times For example if we ip 10 times we have 1024 combinations The reason is that each ip can be either heads or tails and thus each ip doubles our total sequence Or for those of you who are mathematically minded Number of total outcomes 2N where well use the variable N to represent the number of times that were ipping the coin Now this gets very very large very quickly and thus since there is only 1 way to get say all heads then it is very very unlikely to get all heads over a long series of ips Beyond that calculating the exact odds of say 4 heads in a series of 10 ips is a bit tricky and beyond the scope of the course But we can see what the relative probabilities of the di erent outcomes might look like Probability of I 4 6 Number of heads Probability in the Universe Fundamentalsi 1 HNRS 200 Probability in the Universe Lecture 1 Notes You will notice several things about this distribution function First the most likely scenario is that you will get 5 heads 7 exactly half Secondly it is still fairly likely that you well get 4 or 6 or even 3 or 7 heads but fairly unlikely that you will get 01289 or 10 heads But it could happen We describe the average in this case the most likely outcome as the mean and usually label it with the greek letter M mu We call the width of the distribution the standard deviation and usually give it the Greek letter 0 sigma What does the error mean Well a 1 7 0 error means if we did the experiment a gazillion times 68 of the times the result would be in that range Furthermore 95 of the time the result will be within 2 7 lt7 of the average Most scientists use a 2 7 a result as an acceptable standard for an experiment Now you7ll notice something else In addition to a solid distribution there is a smoother looking dotted line which roughly follows the same distribution This is known as a normal distribution and obeys the mathematical relation Jim pm oc e I know It looks ugly That7s why you should just remember what u and 0 mean Now the cool thing is that if you ip lots and lots of times the binomial distribution looks like a normal distribution If instead of a fair coin we have a probability p of coming up heads we nd M Np 1 a iNpu 7 p lt2 which looks complicated unless you plug in p 05 and you see 0 Wz How well does this work Well lets plug in for 50 and 100 coin ips which we7ve already found and Probability of n Probability of n iHHi Eiiii i v i i will will will 0 10 20 30 40 50 0 20 40 so so 100 Number of heads Number of heads Probability in the Universe Fundamentals7 2 HNRS 200 Probability in the Universe Lecture 1 Notes Pretty good eh7 Probability in the Universe Fundamentalse 3 HNRS 200 Probability in the Universe Lecture 2 Notes The Ef cient Market and Random Walks Today we re going to discuss randomness in the context of buying and selling stocks If you ve never played the market don t worry All you need to know to start is that the price of something say a stock is set by being the maximum price people who don t own it are willing to pay to buy it and the minimum price people who do own it are willing sell it for One of the big assumptions in the modern marketplace is that the market is e icient That is information ows through all the buyers and sellers and thus the current price of the stock accurately re ects all the information availa e In principle this is at least approximately testablei We could look at the price of a stock over time and if the market weren t ef cient we could tell A rise in one day for example would more often than otherwise predict a rise the next day and so on While we re not going to do a rigorous statistical analysis consider the following plot of the Dow l ones Industrial Average prices of 30 large stocks over the last century or so wJuIIes Industrial Average ulunu um om am hut kom a M zuu rauu V V om mzn 59 my mum za an1m75 40 cum mm 14anume 1 EB cm 174 7 w my Mom mus 1 WWW mm m 3 115mm M mm 95mm mm 25m az 5n U51u152u253u3540465n556ua 7u75 ua59u95uuu5 It certainly looks like there are clear trends and that a drop in one day or year predicts rop the next year and viceversa But in fact this is a very good example of how your brain can sometimes fool you and detect patterns where there is really noise Consider by contrast the following plot Probability in the Universe The Stock Marketi 1 lINRS 200 Probability in the Universe lecture 2 Notes V lllllllllllllllllll a a 1320 me 1950 WED 2 a This plot was created by doing a random walk Earzh day I ipped a coin well I actually had a computer choose a random number but the effect is the same Since the market tends to go up Imade the coin weighted slightly If it was heads I increased the price by afew percent If it was tails I decreased the price by a few percent Even though in my model every day is completely independent from the last you still see all sorts of regular trends 7 long bull markets a great depression and lots of 5 year business cycles Risk and Return So if the market is random why should we invest at all Why not simply take it to the casino Well for one thing you ll notice that the average Tatum of both the Dow Jones and my model and many stocks by the way are positive over time In fact if you look at the Dow Jones over the last 100 years or so you ll nd that the average return is about 6 per year This is much better than you would normally do by putting your money in the bank However there is atradeoff What would you do if you d found you d invested your money in 1929 and needed it for your retirement 5 years later It s no good telling yourself that the market will eventually go up The scatter in the market 7 known as risk and equivalent to the width of our probability distribution 0 means that occasionally you ll be down over the short term The more risk the longer the period of time when you might be down Economists realize that despite the differences between the tastes of individual investors they share a number of traits in common One of these is that the more risk involved in an investment the more return investors will demand A startup company is very risky and 1A 39 39 39 39 and With aheads 39 39 39 a and atails you walk to Ilse left Probability in the Universe The Stock Market HNRS 2007 Probability in the Universe Lecture 2 Notes thus it should have a high expected return Expected is the keyword7 though7 because even though occasionally you get a stock Which increases by a factor of 50 like Amazon7 you7ll very often get companies like petscorn Which go out of business and Where you7ll lose all of your money Some useful reading 0 A Random Walk dovvn Wall Street777 Burton Malkiel o A Mathematician Plays the Stock Marlltet777 John Allen Paulos Probability in the Universe The Stock Marketi 3 HNRS 200 Probability in the Universe Lecture 9 Notes universe was about as random as you can get Quite literally it complicated than a sea of energy Now you might think th t a newly v nd if there s one thing that I hope ou got ut of q an um mechanics it s anics staggeringly small u y 0 that on small scales randomness is created in the universe fro So around that time if we were to make a map of the universe with hot spots represented by white and cool spots represented by black it might look something like this White Noise stmcmm Linth 5m nil Souls some Does this look familiar It should Basically it s just the white noise that you might see on an olde tyme Tv with an antenna which isn t receiving a c herent signal might call it white noise or static But to a mathematician or a physicist for that matter noise is simply another word for randomness What happens to this noise as the universe evolves and grows 1 As the universe gets larger and larger the radiation gets more and more diffuse and thus it gets cooler w If a hot spot was near a cool spot the two of them might mix and form a lukewarm spot As a similar experiment put an ice cube in a hot drink and see what happens However only nearby spots can mix with one another It takes time to mix and light can only travel at the speed of light Meanwhile the universe is expanding very quickly ome spots are far enough from one another that they might never mix Thus physicists were left with a question If two parts of the universe never mix then why are they very nearly the same This is known as the horizon problem and it can be restated another way Why does one Probability in the Universe Structure in the Universe 1 HNRS 200 Probability in the Universe Lecture 9 Notes of part of the night sky look more or less the same at least on average as any other There are about as many galaxies to the east as to the west for example Why is this especially since as we ll see the hot and cool spots in the early universe were what gave rise to galaxies in the rst place Well around when the universe was about 10 35 seconds old we hypothesize that there was a period of incredibly rapid expansion known as in ation during which the universe increased in size by a factor of about 10100 32 9 a a 391 g E O slt E JAN 9 s o a o 39 3 a q39SM u t Particle Dan Gram LBNL 9 2m Suppomd by DOE and NSF However early in the universe the most important game in town was pressure Believe it or not radiation 7 light 7 can exert enormous forces At early times in the universe the hot and cool spots sloshed back and forth like water waves The biggest waves took the longest to slosh The shortest waves sloshed back and forth very quickly But here s the thing as the universe continued to expand something interested started happening 7 Stuff was being created Photons were colliding with one another creating particles and the particles were colliding to make photons However as the universe cooled the energy of the photons became lower and lower and lower Thus by Einstein s famous equation E m02 1 Eventually there wasn t enough energy to make any new particles and the process stopped Note that now the hot spots become the regions of the universe with the most particles in them The cool spots are relatively devoid of particles Moreover as more and more the radiation in the universe becomes matter the waves slow down stuff typically moves much slower than light and eventually the peaks and valleys become frozen At this point just to give you an update the universe is about 10000 years old There s still a lot of radiation in the universe at this point So much in fact that Hydrogen the most common element in the universe immediately becomes ionized As a result light can t travel very far before running into electrons Thus the universe at that time was very much opaque Probability in the Universe Structure in the Universei 2 HNRS 200 Probability in the Universe Lecture 9 Notes Eventually around when the universe was 100000 years old it was cool enough to form neu tral hydrogen and the light in the universe could freely strearn reaching us today although much much cooler In fact this soup of photons should only be about 3K 3 degrees above absolute zero today and form a background in the universe You can actually detect this with your TV set About 1 of the static on your TV is prirnordial from the creation of the universe In 1964 Penzias amp Wilson discovered the Cosmic Microwave Background or CMB The CMB remains one of the most important rneasurernents of cosrnology because the struc ture we see today arose from the bumps and wiggles within the early universe There are also tiny little bumps and wiggles about 1 part in 100000 within the CMB In 2001 the WMAP satellite was launched to measure the CMB incredibly accurately Here s an image of the creation of the universe And what of those waves in the early universe We can still see them The scales which underwent a full oscillation are still seen as peaks and those which were suppressed are seen as valleys Probability in the Universe Structure in the Universei 3 HNRS 2007 Probability in the Universe Lecture 9 Notes mam m1 rs Cmss inel 3 a m Spwmm mumrim What happened next Say7 for the next 14 Billion years during which the universe evolved Well7 at that point7 the hot spots have become high concentrations of particles7 which7 as I ve said7 attract still more particles One of the things that cosrnologists need to do at this point is model the evolution of the universe using gravity Running typical regions using billions of particles7 we can observe how these initially randorn fluctuations evolve On the course webpage7 you can see a couple of movies which show7 first7 what sirnulations tells us about how the universe evolved7 and secondly7 what we actually see when we look out into the universe Probability in the Universe Structure in the Universei 4