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do any of you guys know the resulting probability distribution from the linear combination of two discrete uniform random variables
- Thread starter RainWaiv
- Start date
Using smogoff to look up answers is something else
this is another ligma set up, isn't it
someone's gonna come in here and say "I discretely distributed in your mom's triangle" right
someone's gonna come in here and say "I discretely distributed in your mom's triangle" right
updog. the answer is updog
mitochondria is the powerhouse of the cell
probability and statistics is fake math i don't associate with those things
Good catch! With a discrete distribution, there's no reason the values need to be sequential. The triangular result is resting on the assumption that the values are equidistant from one another. For example, if you have two random variables A and B, both of which can be any of {1, 2, 6} with probability 1/3, the possible outcomes look like:
2 3 3 4 7 7 8 8 12
Which certainly does not look triangular.
In general I would say that an unspecified discrete uniform variable is likely to have sequential values. But almost everything in stats relies on your assumptions. Technically that's true of all math, but it's much more prevalent in stats. Like, if you're doing linear algebra or something, you don't start every problem by considering whether it's appropriate to model it using vector spaces. The point is, if you're doing stats, you should be cognizant of the assumptions you're making when you apply any rule, formula, etc.
It's nice posting educational material on Smogon. On sites where it's actually on topic, you get spammed with comments like "you're a fucking moron for even trying to teach this person" or "nuh uh, one time in middle school my teacher said birds were a kind of fish, and I trust them more than you"
thats a lot of words
I hate how often this is what Reddit/StackExchange/StackOverflow/etc. devolve into it's such a PITA lol
that plus half its price again
I guess I should be a tinge more specific. I'm only going to be dealing with distributions involving consecutive real numbers, i.e all real numbers within a range from, say, 3 to 7 or 5 to 8.
Consider the case of two identical variables, each ranging from 1 to n. In this situation, the number of possible outcomes is 1/n², each equally likely. Some of those outcomes will be identical. Trivially, we can show that an outcome of 2 or 2n is only possible in one way (both variables being 1 or n, respectively). Likewise, an outcome of 3 or 2n-1 can be made in two ways. We can continue in that pattern until the two sides meet. I'm not going to prove that, because the induction is tricky to set up, and I think you either believe me or else you won't be convinced by induction.
Consider an arbitrary probability distribution. It's easy to see that if we add a fixed value to it, the shape of the distribution does not change, only its position on the x axis.
Now consider two variables, each with the same number of possible outcomes. We'll say variable X ranges from 1 to n, and variable Y ranges from m to m+n. We can rewrite Y as (1 to n)+m. Therefore, X+Y = X+X+m. Using our previous assertion, we can say that the shape of this distribution is the same as the distribution with two identical variables.
If the two variables have a different number of outcomes, the shape will look different. Again I will give a sketch of the proof. Let's say X is a uniform variable from 1 to 4, and Y is a uniform variable from 1 to 20. This is purely in the realm of fantasy; I don't know how you'd generate a uniform random integer from 1 to 20 without some kind of specialized equipment. Here we'll again consider the end behavior first. Still there's only one way to generate a 2, and one way to generate a 24. Likewise, we still have two ways to make 3: 1+2 or 2+1. Note that I'll be writing the combinations as X+Y, so 1+2 would mean that X is 1 and Y is 2. This pattern continues up through 5, which can be generated as 1+4, 2+3, 3+2, or 4+1. Once we reach 6 though, our pattern breaks. 6 can be made as 1+5, 2+4, 3+3, or 4+2. We can't make it with 5+1, because X ranges from 1 to 4. 7 can be made as 1+6, 2+5, 3+4, or 4+3. This pattern again works on both the upper and lower ends, and continues until the two meet. The resulting distribution is trapezoidal. If the smaller distribution has n outcomes, and the larger distribution has m outcomes, we will see 1 combination for the smallest and largest values, 2 for the next values, 3 for the next after that, continuing until the nth value. Afterwards, we will continue to have n combinations until we reach the other end. This makes some intuitive sense: both variables are uniform meaning each outcome on a single variable can be made 1 way. The smaller variable will multiply that by, at most, the total number of outcomes it has. So if the smaller variable has 4 outcomes, we should never expect any result to be made more than 4 ways.
Combinatorics and statistical distributions can get complicated. Even though I'm a math teacher, my formal education is in Computer Science. So before trying to prove stuff, I would honestly suggest playing around with simulations. Any Dice is a great tool for this. It was a big help in convincing myself that my math was right in this post
google wasnt cutting it and i'm not so down bad for information that i'd use ai
plus i know like 30% of smogon members are nerds (do not quote me on this statistic)
