Of course. I was just thinking how different different peoples' idea of "very fast" can be!
Having thought about the problem a bit more, linear interpolation can be done with no divides, if the abscissa spacing of the sample points is uniform. So, just FP adds and multiplies: fast, however you look at it!
For a large number of independent trials, you will find the number of heads has a mean of 500. However, for an individal trial, there is a 32% probability that the number of heads is more than 15 away from the expected mean of 500 (either higher or lower).
If we go to a million flips, there is a 32% probability that the number of heads is at least +/-500 away from the expected mean of 500,000. See how this number has increased, although not as fast as the number of flips?
As your demo applet demonstrates, as N increases the peak gets wider; however, on the plot it appears narrower, since the range of the plot increases faster than the width of the peak. C'mon, this is very basic stuff!
library ( P ) Pronunciation Key (lbrr)
n. pl. libraries
A place in which literary and artistic materials, such as books, periodicals, newspapers, pamphlets, prints, records, and tapes, are kept for reading, reference, or lending.
A collection of such materials, especially when systematically arranged.
A room in a private home for such a collection.
An institution or foundation maintaining such a collection.
A commercial establishment that lends books for a fee.
A series or set of books issued by a publisher.
A collection of recorded data or tapes arranged for ease of use.
A set of things similar to a library in appearance, function, or organization: a library of computer programs.
Genetics. A collection of cloned DNA sequences whose location and identity can be established by mapping the genome of a particular organism.
Divides were (probably still are) so slow that anything that includes one probably can't be very fast:)
Compared to a Taylor's series expansion, which requires a floating-point divide for each term, the single divide in linear interpolation is indeed very fast.
My only hope is that you are purposefully trying to spread misinformation at this point. The only way that the distance is going to grow in either direction is *if* the coin is biased
I'm not trying to spread misinformation, I'm dead serious. As the sample size increases, the standard deviation about the mean grows as SQRT(N)/2. Have a look at the Java applet on this page at the UCB's statistics department; it demonstrates how the difference between the number of heads and the expected number of heads (N/2) increases as N increases. The accompanying text helps explain this. You are clearly the one in the wrong.
I'm a physics student at the U of S (where? everyone asks).
Well, if you were in my class, I'd be failing you. A coin toss sequence is a random walk: at each stage, you have a 50% chance of going forward (heads) or going back (tails). Your total distance travelled is the number of heads tossed at that point. Although the mean distance travelled for a large ensemble of independent random walks is zero, for an individual random walk (or coin toss sequence) this distance tends to grow (in either direction) as the sequence gets longer.
Wouldn't it be nice to work in a field where nobody can say you're wrong?"
Unfortunately, the postmodernists have attempted to apply their idiotic claptrap to science, claiming the existence of such absurd concepts as "alternative scientific truths". What they miss is that science is empirical, and therefore deals with observed characteristics of the real world (i.e., "facts").
I've always wanted to throw one out of a plane over China, and yell after them as they plummet to their death: "how are you finding that Far-Eastern Gravitation?"
I don't know which method is generally accepted as best for computer calculations
Taylor's series approximations are notoriously bad at converging to trig functions; you need many terms to even approach the true value, and that makes a series a bad choice for fast calculations.
Using complex exponentials is just pushing the burden off onto the exp() function, and there is also the overhead of complex arithmetic.
In fact, many modern high-performance compilers use look-up tables for trig functions, combined with some form of interpolation. For linear interpolation, you need a*log_2 (N) operations to find the correct position in the table (where N is the table size, and a is a small constant), and then 4 floating-point adds, 2 multipies, and 1 divide. Very fast!
So you see that the peak is relatively smaller for more samples. As far as the 'absolute terms', I'm not sure what your point is.
My point is exactly what you stated: the peak gets relatively narrower as N gets larger, but the standard deviation becomes larger when considered on its own (i.e., without reference to the mean). This is actually quite an important result, since it explains how random walk processes (where the mean position is zero) can lead to physical phenomena such as diffusion. Feel stupid now?
It's a pity that the present-day language of choice for high-performance computing, Fortran 90/95/HPF, was not covered in this study. There has been anecdotal evidence that C++ has approached Fortran, performance-wise, in recent years, but I've yet to see a proper comparison of the two languages.
I'm sorry, you are wrong here. The correct expression is sigma x-bar = sigma * sqrt(n)/ sqrt(n-1). With this mistake in your statistics, all of your conclusions are incorrect.
The variance for the number of heads is N/4 so the standard deviation is Sqrt(N)/2.
Oops, I got the numerical factor wrong. But the important point is that the standard deviation increases as N increases.
I don't know what "absolute terms", "relative terms" or "law of averages"
In absolute terms, the standard deviation is just Sqrt(N)/2 as you state; this value increases with N. However, this value as a fraction of the mean number of heads (N/2) varies as 1./Sqrt(N), which decreases with N.
any sum of N samples that each follow the Bernoulli distribution with p = 1/2 will have a sample mean that approaches N/2 as N tends to infinity
...and that's were you are wrong. If the standard deviation is growing without bound, then the chances of being anwhere near N/2 (like, a small number of heads off) for large N is pretty small. However, the chances of being within N/2+SQRT(N)/2 and N/2-SQRT(N)/2 are much better, in fact around 68% (IIRC) for a two-sided normal distribution out to 1 sigma.
The 'decrease in biodiversity' has been going on for as long as man has walked on two feet.
Not at the current rate, it hasn't. You offer some anecdotal evidence about horses being wiped out in North America by Native Americans (which I don't find inconceivable), but this has scant relevance to the present debate. What has far more relevance is the mounting evidence that Earth may be experiencing one of the largest mass extinctions of all time. This has not been going on since Man first walked on two feet.
A thread of anti-modernism runs deep within many intellectual circles
Judging from the quality of your rhetoric, I'm amazed at your familiarity with these "intellectual circles".
nasa has a plan for a lander on europa complete with a sub-ice probe that's been sitting on the backburner for years.
I wouldn't even call these plans; at the moment, the only Europa-relevant mission currently under consideration by NASA is the Jupiter Icy Moons Orbiter (JIMO). Unfortunately, as its name implies, JIMO won't have a lander facility. The mission, if it goes ahead, will be launched no sooner than 2011.
I would vote for the capitalist, as he actually owns the plant and is forced to react to market preasures
Ah, and there's the rub. The capitalist only reacts to market pressures, not through any motivation to help his fellow man. If it were (a) legal to torture babies by stubbing out cigarettes into their eyes, and (b) profitable, then the capitalist would have no recourse but to do it; after all, he has to react to market pressures and think about his bottom line.
Remember: these are the same market pressures which have led to the wholesale dumbing down of television in the US.
Animals that have trouble with the new enviroment will simply migrate.
Unfortunately, the plant species which lie at the bottom of the food chain can only migrate at rate of about 10m/year. Which just ain't fast enough to deal with changes happening over timescales as small as decades. Accordingly, the animals can migrate all they want, but there won't be anything for them to eat when they've moved...
but many hard core enviro-freaks are also die hard socialists.
And this is such a bad thing? Consider a choice: on the one hand, a capitalist who doesn't want to dump mercury in the water supply because it will incurr a fine and hit his bottom line; and on the other, a socialist who doesn't want to dump mercury in the water supply because he doesn't want to poison the organisms (including humans) who rely on that supply. My vote in this situation would go to the socialist; how about you?
And you can't tell me whether this coin I flip is going to come up heads or tails, but claim that in if I do it 1000 times, I will get very close to 500 heads? Bah!
In 1000 tosses, you probably won't get close to 500 heads. This is because the standard deviation of the head count for N tosses varies as SQRT(N/2), centred on the mean of N/2. In absolute terms, therefore, more tosses will get you *further* from the current mean. Only in relative terms will more tosses get you closer. This is why the so-called 'law of averages' is wrong.
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Of course. I was just thinking how different different peoples' idea of "very fast" can be!
Having thought about the problem a bit more, linear interpolation can be done with no divides, if the abscissa spacing of the sample points is uniform. So, just FP adds and multiplies: fast, however you look at it!
For a large number of independent trials, you will find the number of heads has a mean of 500. However, for an individal trial, there is a 32% probability that the number of heads is more than 15 away from the expected mean of 500 (either higher or lower).
If we go to a million flips, there is a 32% probability that the number of heads is at least +/-500 away from the expected mean of 500,000. See how this number has increased, although not as fast as the number of flips?
As your demo applet demonstrates, as N increases the peak gets wider; however, on the plot it appears narrower, since the range of the plot increases faster than the width of the peak. C'mon, this is very basic stuff!
library ( P ) Pronunciation Key (lbrr) n. pl. libraries
Divides were (probably still are) so slow that anything that includes one probably can't be very fast :)
Compared to a Taylor's series expansion, which requires a floating-point divide for each term, the single divide in linear interpolation is indeed very fast.
My only hope is that you are purposefully trying to spread misinformation at this point. The only way that the distance is going to grow in either direction is *if* the coin is biased
I'm not trying to spread misinformation, I'm dead serious. As the sample size increases, the standard deviation about the mean grows as SQRT(N)/2. Have a look at the Java applet on this page at the UCB's statistics department; it demonstrates how the difference between the number of heads and the expected number of heads (N/2) increases as N increases. The accompanying text helps explain this. You are clearly the one in the wrong.
I'm a physics student at the U of S (where? everyone asks).
Well, if you were in my class, I'd be failing you. A coin toss sequence is a random walk: at each stage, you have a 50% chance of going forward (heads) or going back (tails). Your total distance travelled is the number of heads tossed at that point. Although the mean distance travelled for a large ensemble of independent random walks is zero, for an individual random walk (or coin toss sequence) this distance tends to grow (in either direction) as the sequence gets longer.
At the heart of deconstruction is the theory that know understands what anyone else is saying
So I guess your post was actually a postmodern?
Wouldn't it be nice to work in a field where nobody can say you're wrong?"
Unfortunately, the postmodernists have attempted to apply their idiotic claptrap to science, claiming the existence of such absurd concepts as "alternative scientific truths". What they miss is that science is empirical, and therefore deals with observed characteristics of the real world (i.e., "facts").
I've always wanted to throw one out of a plane over China, and yell after them as they plummet to their death: "how are you finding that Far-Eastern Gravitation?"
I don't know which method is generally accepted as best for computer calculations
Taylor's series approximations are notoriously bad at converging to trig functions; you need many terms to even approach the true value, and that makes a series a bad choice for fast calculations.
Using complex exponentials is just pushing the burden off onto the exp() function, and there is also the overhead of complex arithmetic.
In fact, many modern high-performance compilers use look-up tables for trig functions, combined with some form of interpolation. For linear interpolation, you need a*log_2 (N) operations to find the correct position in the table (where N is the table size, and a is a small constant), and then 4 floating-point adds, 2 multipies, and 1 divide. Very fast!
So you see that the peak is relatively smaller for more samples. As far as the 'absolute terms', I'm not sure what your point is.
My point is exactly what you stated: the peak gets relatively narrower as N gets larger, but the standard deviation becomes larger when considered on its own (i.e., without reference to the mean). This is actually quite an important result, since it explains how random walk processes (where the mean position is zero) can lead to physical phenomena such as diffusion. Feel stupid now?
It's a pity that the present-day language of choice for high-performance computing, Fortran 90/95/HPF, was not covered in this study. There has been anecdotal evidence that C++ has approached Fortran, performance-wise, in recent years, but I've yet to see a proper comparison of the two languages.
Whether millions of years are needed depends on your definition of "evolution"
From your Canada Goose example, you obviously don't understand what evolution means. Therefore, any further discussion is fruitless.
sigma x-bar = sigma / sqrt(n)
I'm sorry, you are wrong here. The correct expression is sigma x-bar = sigma * sqrt(n)/ sqrt(n-1). With this mistake in your statistics, all of your conclusions are incorrect.
The variance for the number of heads is N/4 so the standard deviation is Sqrt(N)/2.
Oops, I got the numerical factor wrong. But the important point is that the standard deviation increases as N increases.
I don't know what "absolute terms", "relative terms" or "law of averages"
In absolute terms, the standard deviation is just Sqrt(N)/2 as you state; this value increases with N. However, this value as a fraction of the mean number of heads (N/2) varies as 1./Sqrt(N), which decreases with N.
any sum of N samples that each follow the Bernoulli distribution with p = 1/2 will have a sample mean that approaches N/2 as N tends to infinity
...and that's were you are wrong. If the standard deviation is growing without bound, then the chances of being anwhere near N/2 (like, a small number of heads off) for large N is pretty small. However, the chances of being within N/2+SQRT(N)/2 and N/2-SQRT(N)/2 are much better, in fact around 68% (IIRC) for a two-sided normal distribution out to 1 sigma.
...Bush will announce plans for two moonbases: Moonbase Alpha and Moonbase Zappa.
(with apologies to Mike Myers)
Nope, but nice ASCII art. I'm curious to know what you are getting at?
The 'decrease in biodiversity' has been going on for as long as man has walked on two feet.
Not at the current rate, it hasn't. You offer some anecdotal evidence about horses being wiped out in North America by Native Americans (which I don't find inconceivable), but this has scant relevance to the present debate. What has far more relevance is the mounting evidence that Earth may be experiencing one of the largest mass extinctions of all time. This has not been going on since Man first walked on two feet.
A thread of anti-modernism runs deep within many intellectual circles
Judging from the quality of your rhetoric, I'm amazed at your familiarity with these "intellectual circles".
using conventional rockets, a mars trip would take at least 2 years
But Mars Express got there in 6 months. How is that?
nasa has a plan for a lander on europa complete with a sub-ice probe that's been sitting on the backburner for years.
I wouldn't even call these plans; at the moment, the only Europa-relevant mission currently under consideration by NASA is the Jupiter Icy Moons Orbiter (JIMO). Unfortunately, as its name implies, JIMO won't have a lander facility. The mission, if it goes ahead, will be launched no sooner than 2011.
Maybe the Pentagon saw a turban poking up behind a boulder in one of the Sprit MER pictures?
I would vote for the capitalist, as he actually owns the plant and is forced to react to market preasures
Ah, and there's the rub. The capitalist only reacts to market pressures, not through any motivation to help his fellow man. If it were (a) legal to torture babies by stubbing out cigarettes into their eyes, and (b) profitable, then the capitalist would have no recourse but to do it; after all, he has to react to market pressures and think about his bottom line.
Remember: these are the same market pressures which have led to the wholesale dumbing down of television in the US.
Animals that have trouble with the new enviroment will simply migrate.
Unfortunately, the plant species which lie at the bottom of the food chain can only migrate at rate of about 10m/year. Which just ain't fast enough to deal with changes happening over timescales as small as decades. Accordingly, the animals can migrate all they want, but there won't be anything for them to eat when they've moved...
but many hard core enviro-freaks are also die hard socialists.
And this is such a bad thing? Consider a choice: on the one hand, a capitalist who doesn't want to dump mercury in the water supply because it will incurr a fine and hit his bottom line; and on the other, a socialist who doesn't want to dump mercury in the water supply because he doesn't want to poison the organisms (including humans) who rely on that supply. My vote in this situation would go to the socialist; how about you?
And you can't tell me whether this coin I flip is going to come up heads or tails, but claim that in if I do it 1000 times, I will get very close to 500 heads? Bah!
In 1000 tosses, you probably won't get close to 500 heads. This is because the standard deviation of the head count for N tosses varies as SQRT(N/2), centred on the mean of N/2. In absolute terms, therefore, more tosses will get you *further* from the current mean. Only in relative terms will more tosses get you closer. This is why the so-called 'law of averages' is wrong.