This doesn't seem surprising me and at least intuitively they represent the same logic that you can proof that with some substitutions. Specifically as soon as you start thinking of the square root as the product of two random numbers rather than a single number it shifts your perception, assuming the random numbers all fall between 0 and 1 with the same distribution.
I am taking probability class this semester, so I am finally able to understand this as a continuous random variable and I'm still surprised that the CDF is x²
So "both of these processes have the same cumulative distribution function, that's why they are identical." No, that's the definition of identical, not the why. (How else did you even know that they were equal?? You just said so, with no justification, at (0:40) .. "in the sense that the distribution describing these results is the same.") As you said, the probability of a specific outcome like 0.7 is zero. What's the same for both is the prob (X < 0.7). That's the distribution.
No. It should not feel "wrong" that two different random processes have the same probability. There's no reason one thing cannot be the result of two separate processes. (To give a simplified example, "13" is both the answer to "6 + 7" and also "26 divided by 2." Is that "wrong" ?? No. Likewise with x^2 here.)
It makes sense as the limit of the number of terms approaches infinity too. Suppose you took the max of infinite randoms between 0 and 1–you probably get 1; and, if you take some number to a power near zero, you also get one. Pretty cool.
Well it's not identical because imagine two 0->1 numbers n1 and n2. the probability of max(n1,n2) being larger than the initial numbers n1 and n2 is 1-n1 for n2 and 1-n1 for n1. square root's uses 1 number n1 and ALWAYS scales it up, probability of 100. that's like saying f(x)=x is steeper than f(x)².
Like every probability proof, I cant tell if this is trivially easy – or the proof is so clever as to make it obvious. I went through a math of probability textbook, and it felt like I just rewrote the normal distribution 200 times.
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Yes, and can be generalized to higher dimensions !
I actually used this when generating a random number with the distribution leaning more to the top.
Seems like this would help solve the three body problem.
calculate max(rand rand) using sqrt(rand) – nah
calculating sqrt(rand) by max(rand rand) – absolute gigabrain
How to calculate the possibility of a successful Windows 11 update…
What does computing the maximum of two numbers mean?
I think second expresion should be “rand(max_of_rand_range)”
This doesn't seem surprising me and at least intuitively they represent the same logic that you can proof that with some substitutions. Specifically as soon as you start thinking of the square root as the product of two random numbers rather than a single number it shifts your perception, assuming the random numbers all fall between 0 and 1 with the same distribution.
Do it be sqrtin?
Let me guess, it’s about Pi? 🙄
It faster to get 2 random numbers than doing a square root out of one random number.
From the fact that we visualized PoE lucky graphs so many times, it sounds about right, but how the heck…
Dear 3blue1brown,
I am taking probability class this semester, so I am finally able to understand this as a continuous random variable and I'm still surprised that the CDF is x²
Thanks to this video, I'mma start calling square root 'squirt'
So "both of these processes have the same cumulative distribution function, that's why they are identical." No, that's the definition of identical, not the why. (How else did you even know that they were equal?? You just said so, with no justification, at (0:40) .. "in the sense that the distribution describing these results is the same.") As you said, the probability of a specific outcome like 0.7 is zero. What's the same for both is the prob (X < 0.7). That's the distribution.
No. It should not feel "wrong" that two different random processes have the same probability. There's no reason one thing cannot be the result of two separate processes. (To give a simplified example, "13" is both the answer to "6 + 7" and also "26 divided by 2." Is that "wrong" ?? No. Likewise with x^2 here.)
Yeah I got this problem with density function wrong on an exam a long time ago because I didn't know this
THAT. IS. BEAUTIFUL.
this is cool af
I'll stop watching your short-form when you start making long-form again, Mr. Sanderson.
Very intuitive to Dnd players who like math
Or said another way, the number of variables play power mechanics in determining distributions of their functions ?
It makes sense as the limit of the number of terms approaches infinity too. Suppose you took the max of infinite randoms between 0 and 1–you probably get 1; and, if you take some number to a power near zero, you also get one. Pretty cool.
Randomness bringing algorithms together!
Well it's not identical because imagine two 0->1 numbers n1 and n2. the probability of max(n1,n2) being larger than the initial numbers n1 and n2 is 1-n1 for n2 and 1-n1 for n1. square root's uses 1 number n1 and ALWAYS scales it up, probability of 100. that's like saying f(x)=x is steeper than f(x)².
Lots of random variables have the same distribution, for example Normal distribution. This must happen a lot.
Like every probability proof, I cant tell if this is trivially easy – or the proof is so clever as to make it obvious. I went through a math of probability textbook, and it felt like I just rewrote the normal distribution 200 times.