This is really a bad title, but random numbers indeed need not be uniformly distributed - for example Gauss distributed numbers aren't distributed uniformly (from definition:).
If the random variable that we measure has Gaussian distribution - the numbers we read will have Gaussian distrubuition, and will be random.
Other example - rolling dice with 5 sides signed "6" and one side signed "1" - results will be random, but the resulting distribution won't be uniform.
Uniform distribution is special case of random distribution.
Ah. Fair enough. That's a rather interesting case, and what I assume is not what people think of when you say random number. I assume that people think of a uniform distribution (i.e equally likely numbers).
You're probably right that people who haven't taken a course in probability are likely to use "random" to mean "uniformly at random". And of course, most people haven't taken a course in probability.
I would assert that people don't even know how to conceptualize the question in these terms ("random number" == "uniform distribution").
After you do study probability (by gambling, by lab work, by quantitative programming, or, least fun, in coursework), you get introduced to the "right questions to ask", like:
"what's the distribution",
"are the samples independent",
"what are you conditioning upon"
For instance, many people don't know that the distribution of the sum of two dice looks very different than just one die. They're both random, but not both uniform. Often people know something's weird, but can't formulate the question because they don't have the language.
Speak for yourself. Gaussian distributions are ubiquitous (thanks to the Central Limit Theorem), while uniform distributions only exist in the realm of pseudo-number generators and little else.
I think that the point you're trying to make is that a random variable with a uniform distribution is more "random" than a random variable with a gaussian distribution. The metric you're implying is called entropy, as a uniform distribution is maximally entropic.
I thought the sum of several dice rolls was a Gaussian distribution to start with. What more do you need just roll say 10 dice and then pick a threshold value?
Exactly. 3d6 is already pretty close, and it just gets closer and closer to the gaussian as you increase the number of dice. And it's bounded, too, so you get minimum and maximum values.
Comments
This is really a bad title, but random numbers indeed need not be uniformly distributed - for example Gauss distributed numbers aren't distributed uniformly (from definition:).
I don't understand in what sense numbers drawn from a Gaussian distribution would be considered random.
If the random variable that we measure has Gaussian distribution - the numbers we read will have Gaussian distrubuition, and will be random.
Other example - rolling dice with 5 sides signed "6" and one side signed "1" - results will be random, but the resulting distribution won't be uniform.
Uniform distribution is special case of random distribution.
Ah. Fair enough. That's a rather interesting case, and what I assume is not what people think of when you say random number. I assume that people think of a uniform distribution (i.e equally likely numbers).
You're probably right that people who haven't taken a course in probability are likely to use "random" to mean "uniformly at random". And of course, most people haven't taken a course in probability.
I would assert that people don't even know how to conceptualize the question in these terms ("random number" == "uniform distribution").
After you do study probability (by gambling, by lab work, by quantitative programming, or, least fun, in coursework), you get introduced to the "right questions to ask", like:
"what's the distribution", "are the samples independent", "what are you conditioning upon"
For instance, many people don't know that the distribution of the sum of two dice looks very different than just one die. They're both random, but not both uniform. Often people know something's weird, but can't formulate the question because they don't have the language.
Speak for yourself. Gaussian distributions are ubiquitous (thanks to the Central Limit Theorem), while uniform distributions only exist in the realm of pseudo-number generators and little else.
I think that the point you're trying to make is that a random variable with a uniform distribution is more "random" than a random variable with a gaussian distribution. The metric you're implying is called entropy, as a uniform distribution is maximally entropic.
I've actually looked at ways of combining die rolls so that there are an x and y variable applied to them and they produce as Gaussian distribution:
Increasing the x would decrease the range of the distribution, while increasing the y would increase the magnitude of the distribution.
I never found something simple enough for what I wanted to apply it to (a game).
I thought the sum of several dice rolls was a Gaussian distribution to start with. What more do you need just roll say 10 dice and then pick a threshold value?
Exactly. 3d6 is already pretty close, and it just gets closer and closer to the gaussian as you increase the number of dice. And it's bounded, too, so you get minimum and maximum values.
"Uniform distribution is special case of random distribution."
What is a random distribution? I think you meant probability distribution.
Indeed - sorry - my math Enlish isn't very good.
I don't understand in what sense numbers drawn from a uniform distribution would be considered random.
Do you think the result of rolling two dice and adding them is random?
The numbers' only relationship is to the distribution, not each other or some other process.