Source
r/math•
Jengu

Newb question about randomness

If I generate a bunch of random numbers and plot them, I'll see a bell curve. That I can predict that suggests it's nonrandom. Why do random numbers make predictable graphs? Wouldn't it be more random to always get back a different graph? Is there a different term for that 'kind' of randomness?

Edit: To give some background for explanations, I've taken calculus but never statistics and know that generated 'random' numbers aren't really random. But the less background needed to understand a response the better, for other redditors sake.

Edit edit: Sorry, I indeed did mean if you plot the distribution of the numbers. Some have pointed out that this is a characteristic of a particular random number generator's implementation. Why would you ever want this type of distribution for random numbers? Would a distribution resulting in a flat line be 'more' random? (or as close to a flat line as possible since we can't have true randomness -- a uniform distribution seems more nonrandom than we have to be)

Edit edit edit: ahahaha, just to add to the confusion I understood "uniform" and "normal" distributions to be the same when writing some of my replies. facepalm

Posted Jun 21, 2010, 5:11 PM UTC · edited

247
punkmonk•

"If I generate a bunch of random numbers and plot them, I'll see a bell curve."

First, I think you mean plot their distribution histogram. That is, on x axis you have the value, and y axis, the frequency.

Second, you will only see a bell curve, if the numbers you are generating come from a Gaussian distribution

If you generate numbers from a uniform distribution, you will see a flat line

Numbers can come from any distribution. Gaussian distribution (a.k.a Normal distribution or bell curve distribtuion) is just one of them.

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JenguOP•

That is what I meant, see my edit.

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[deleted]•

Poisson, Uniform, Beta-distributed, Geometric, Binomial . . .

Delicious!

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diffyQ•

I'm sorry that this correct answer isn't closer to the top. To paraphrase, you will see a bell-curve-shaped histogram if you are sampling from a distribution with a bell-curve-shaped density (or mass function). Sounds like the OP is confused by the statement of the central limit theorem.

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cavedave•

A pseudo random number generator with the same initial seed number should always return the same sequence of 'random' numbers. Random numbers should not follow a bell curve distribution. Are you random number the addition of two random numbers? If you take two dice and roll them repeatedly the numbers that come up will follow a bell curve distribution. The numbers are still random but adding them makes the distribution non random.

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Avagad•

Random numbers could follow a bell curve distribution. If I go out of the house and measure people's height as I meet them then their height in cm (or whatever) is random but if I were to graph it then it would (probably) follow a bell curve distribution. The only requirement to a random number is surely that the next number in the sequence has no dependency on any of the previous numbers.

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cavedave•

Heights do follow a bell curve distribution roughly. Women and men are two different curves. North and south Koreans are also two different ones. Pigmys and Massai are also.

But height is not really random is it? The probability of meeting an adult 1 cm tall or 5000 centimeters tall is fairly low. Height of person N has no effect on the height of person n+1 they are independent of each other.

Having measured a number of heights I can with good confidence

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Avagad•

But height is not really random is it? The probability of meeting an adult 1 cm tall or 5000 centimeters tall is fairly low. Height of person N has no effect on the height of person n+1 they are independent of each other.

But there is no dependency on distribution in the production of random numbers. Rolling a d20 with all the sides set to 1 except one with 20 is still random as you roll it despite the fact you'll probably get 1.

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cavedave•

Imagine all the sides were 1. You are claiming that a sequence of 1 repeatedly is random? I agree the side landed on with a dice is random but not a series of n 1's with a program complexity of log n

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rockinchizel•

It depends on how you are generating the random numbers. If you instruct the computer to pick random reals from zero to infinite, and then plotted the number of times each number was picked, you would get something much closer to the line y = 1. Random numbers don't make predictable graphs, the distribution of random numbers make predictable graphs. If generated our random numbers by picking random people who took the SAT and looking at their SAT Math Score, we would only generate numbers between 200 and 800. However, we don't see the curve appear until we start looking at how many times we pull something like a 500, or a 600, and then when we plot (500, number of times we pull a 500), for all possible scores, then we start to see the bell curve.

If our graph of frequencies is a horizontal line, that is a good thing because it means that each number is just as likely to occur as any other, and thus we have true randomness. An example of something like this would be to have you type a 10 letter word, and then to record the time your key stroke falls down to the 6th decimal place of the second, and then to make a number that is just all of those 6th decimal places lined up. Since we don't have the motor controls to effectively time our typing down to the 6th decimal place of the second, a number generated in such a way would be about as random as it could be.

I think the thing you should take away from this is that what you think of as a predictable graph is not generated by the random numbers themselves, but rather by the frequencies in which each number is chosen. A truly random number generator would have a very predictable graph (a horizontal line) because all possible outcomes would be equally likely.

I hope this has clarified some things for you.

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zair•

One quibble: you say a horizontal line suggests 'true randomness'. However, the sequence 01234567890123... will produce said horizontal line but is in fact highly predictable. So, even though I understand your meaning, calling it 'true randomness' is incorrect.

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rockinchizel•

Good point. But, while it only suggests true randomness, there have to be other indicators there as well. Such as no pattern. But yes, we cannot base our definition of randomness solely on the distribution graph.

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Avagad•

true randomness

"Randomness" has nothing to do with the distribution of the numbers.

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JenguOP•

Why not pick algorithms that usually result in something closer to a horizontal line? A uniform distribution looks gratuitously nonrandom.

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jeremybub•

Essentially, a "bell shaped distribution", which means a lot of the samples are average, and few are extremely big or small, fits a lot of naturally occuring data. Look at human height for example. A lot of people are average height, and as you look at taller and taller people, there are fewer and fewer of them. Same with short people. Thus we end up with this "hump" around the average value, that tails off, or the "bell" shape.

It's really about matching whatever you are trying to emulate. If you flip 100 coins, and count the number of heads, you would expect it to be around 50. However, sometimes it would be more, and sometimes it would be less. This is the "bell shaped" distribution we were talking about. A "gaussian" random number generator, or one that generates a "bell shaped curve", would approximate this extremely well.

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JenguOP•

It's really about matching whatever you are trying to emulate. If you flip 100 coins, and count the number of heads, you would expect it to be around 50. However, sometimes it would be more, and sometimes it would be less. This is the "bell shaped" distribution we were talking about.

I think you're confusing levels? Isn't sometimes getting more than 50 and sometimes getting less than 50 looking at the distribution of the distribution, i.e. a trial of trials?

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jpfed•

Not necessarily- it sounds like he's just looking at it from the perspective of a process that generates a binomial distribution instead of 100 trials from a Bernoulli distribution.

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jeremybub•

Well I'm looking at the probability as a whole of the entire 50 coin run. Like if that were a game, and you got a dollar for each heads, and you wanted to figure out what you are likely to win.

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rockinchizel•

Why? What is so non-random about being just as likely to choose one number over another? It would be decidedly non random if you had an upward sloping curve, because then we would be more likely to get higher numbers. Or if you had a sine curve, we would be more likely to pick numbers that are inputs where sine has a peak. I guess I don't understand why you expect that the graph of the distribution would look "random." We obviously have to have a function, as we can't get x=1 both 2 and 5 times. Thus, we couldn't have like a scatterplot that is freckled by a bunch of "random" dots. Why do you believe that the distribution of a random number generator should look random as well?

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JenguOP•

What is so non-random about being just as likely to choose one number over another?

It's not just as likely, that's the issue. A guassian random number generator picks numbers in the middle more often, that's why the distribution looks like a bell curve.

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rockinchizel•

I was referring to your previous statement that "a uniform distribution looks gratuitously nonrandom." I agree that you are less likely to get numbers at the tail end of a Gaussian distribution. But if your random number generator generates numbers that have a Gaussian distribution of frequencies, you have a shitty random number generator. Gaussian distributions come from data about populations, they are not used to generate random numbers

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Enginerd•

Lol, that's "randomness for you". There are gaps and voids. Algorithms do exist called "quasi-random" number generators, not to be confused with "pseudo-random" number generators. A "quasi-random" number generator attempts to mimic an even distribution in aggregate (a horizontal line in this case).

As for "why" use one, it depends on the application. For cryptography, you want to be as random as possible. For scientific simulation, you sometimes want quasi-random.

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RegularGuest•

I would say the sequence of numbers generated that determines your distribution (bell-curve in your example) are different each time.

Say you have a discrete uniform distribution across the first 10 positive integers (ie the probability of picking 1, 2, 3 etc = 1/10). Over time you would expect the number of 1's 2's 3's etc. generated to be equal. However, the order of numbers picked to arrive at this distribution isn't necessarily the same.

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punkmonk•

Entropy is a classical measure of randomness. It measures expected number of bits you need to represent the random number.

For example, for a fair dice roll, you cannot make any assumption on what will the next value be. So you can't use any clever compression teqniues to save a sequence of dice rolls.

However, if you have a crooked dice, and you 50% of the times you get a "3". The maybe you can represent three with a smaller code than others. Hence achieving good compression. This dice is less random, and has lower entropy.

Uniform distributions have highest entropy. So yes, flat line or uniform distirbutions can be considered more random than bell-curve distributions.

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goldenflaw•

When I think of a random number without any context, I'm thinking of a number not more likely than any other. Me being able to predict that the distribution is going to be uniform means precisely that I have no good guess about what the next number is going to be which is what I expect.

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[deleted]•

Why do random numbers make predictable graphs? Wouldn't it be more random to always get back a different graph?

It seems to me that this is the crux of your question. The problem is, your question assumes that there can be a random variable which always shows a different, unpredictable distribution, but the truth is that this is not easy or perhaps even possible.

It may be clearer to think about a simpler concept. Suppose you generate a bunch of random numbers and find their average. Do you think you'll see wildly different averages every time you do this? What if you generated a million random numbers and averaged them? Do you think the result would be totally unpredictable, or do you think you could make an accurate guess? Does the law of large numbers make sense to you? The fact that given enough samples, you recover the underlying distribution of the random variable is a sort of generalization of the law of large numbers.

Perhaps the randomness you're thinking of is something like the weather, where if you sample the temperature, wind speed or other meteorological variables at random times, the distribution you'd get wouldn't be the same at different days. But on a long enough timescale, even those fluctuations get smeared out and you get the predictable, stable distribution of weather that we call climate. Or perhaps you're imagining thinking up wildly varying random numbers in your head and producing a different crazy scattering each time. But accumulate that over a million trials, and even there a consistent distribution will emerge.

Besides, even if you have perfect knowledge of the distribution of a random process, it tells you nothing about what particular random numbers will come out of it. I know that the roll of a die is equally likely to come out any number between 1 and 6, but that doesn't help me cheat at board games. A uniform distribution doesn't mean that if I've the last rolls have been lots of ones and twos and no sixes then the next roll is likely to be a six: the "law of averages" isn't the same as the law of large numbers.

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[deleted]•

[deleted]

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JenguOP•

But we're just talking about plotting the frequency. There's no summing involved...

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[deleted]•

Just to clarify: your situation is basically you're generating numbers in some range (say [0,1]), and then plotting the number of numbers that were generated within small intervals (say [0,0.1), [0.1, 0.2) and so on). And you're saying that gives you something that looks like a normal distribution?

If that's the case, then either you're using a horrible random number generator, or one that is designed to generate numbers from a normal distribution.

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efrique•

It sounds like you are confusing two different notions of randomness, one intuitive and one probabilistic, only one of which is relevant to generating random numbers.

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sawu•

I know there are now quite a lot of replies, and a lot of the discussions have veered away from your original question. So I'm going to try summarise this:

In your method of generating random numbers, the underlying system will be something like getRandom(). Basically, this is a psuedo-random number generator. It essentially has a seqeuence defined, that will return numbers based on a certain distribution.

It seems as though the one you are using, uses the normal/gaussian distribution to generate the numbers. When plotted, you will indeed see a bell curve.

These distributions give you a probability of generating a number in a certain range. For example the probabilty of finding 0.25<x<0.75 might be a half.

Now, this has many uses, from estimating statistics (you have 300 people, with a mean height of 1.6 and a standard deviation of 0.3, you can calculate the probability that a given person will be X metres tall since height generally satisfies a normal distribution)

So this randomness is predictable to an extent, sure.

With the uniform distribution (i.e. the distribution is a 'straight line') we have the exact same thing. However the probability of a randomly generated number being in a certain range is constant.

If numbers are generated from 0 to 1, then the probability that a number is in the range 1/a<x<1/b is precisely 1/b-1/a. (where a,b are from 1,2,3,....)

This also has it's uses. If all you wish to return is a random non-biased number, then you would use the uniform distribution.

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Enginerd•

Let's say by "random number" you mean the output will be:

  1. An integer between 0 and 2^M -1.
  2. Equally likely to be any of the numbers chosen in step #1.
  3. Independent of any of the other trials.

So if we have a sequence of numbers S(n), then we know that 0<=S(n)<2^M, but knowing S(n-ii) for any value of ii will not help you figure out S(n) (other than the trivial ii = -1).

You can't figure out much about what a given number in the sequence will be, if you don't already know it. But you can figure out the distribution of aggregates. The uniform number example isn't too interesting, but you can figure out what the standard deviation of the sequence is likely to be, and you can get a distribution of the average after

What you can figure out are the odds of getting aggregate results after N trials. So to answer your question, the aggregates statistics aren't random, but they aren't certain. You say "you'll get a bell curve", but what's actually true is that the probability of you getting particular results is extremely high.

I guess the summary is that just because a process contains randomness doesn't mean you can't figure anything out about it at all.

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mmc31Probability•

Something you may find interesting then is the benford distribution. If you open a book and pick the first digit at random that you find, it will not be distributed uniformly between 0 and 9. It will be distributed according to the benford distribution, and will be 1 almost 1/3 of the time.

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[deleted]•

[deleted]

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NanoStuff•

Hmm, no. Seeing a normal distribution is definitely not normal, pardon the pun.

Arithmetic means of random numbers, or just sums of random numbers, will produce it, but a plain old random number generator should not do this on it's own. The numbers should be evenly distributed.

He should be more specific regarding how he generated those numbers, but an old fashion rand() function should not do that.

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JenguOP•

Updated with more specificness.

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NanoStuff•

For you question, why would anyone want a normal distribution, there are more reasons that I can think of. Boost has that as a built in random function because it's just so darn useful.

If you want randomness when shooting a gun in a game (so bullets dont hit same exact spot), you dont want the hits to be uniformly distributed over the entire space, or uniformly distributed about some limited range and then abruptly cut off. You want a normal distribution around the center.

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JenguOP•

But it's deliberately less random! Seems like either a bad default or a poorly named function.

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[deleted]•

[deleted]

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NanoStuff•

I didn't downvote you, for the record :) And you're right of course, the type of distribution is of no consequence to the randomness of the numbers. A normal distribution is just as random as the numbers which created it.

I am still curious how he is getting the normal distribution however, as such a thing should definitely not come out of a basic random number generator.

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billmeyersriggs•

I'm confused as well - to me, the tools most easily available to a "newb" would produce a uniform distribution.

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TBcasualty•

You're talking about PSEUDORANDOM NUMBERS. You can't really tell a computer to do something random, it has to follow strict guidelines. For example, you can give it a seed, which is a number that in some way incurs another number. So, maybe that number will be multiplied by the current milisecond held in the computer's internal clock, and will then be returned modulo the desired range.

Does that help?

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JenguOP•

No, because if you read my OP you'd see I already knew this.

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