Interesting, it seems like x,y,z are independent samples which is why we can multiply their 3 respective uniform probabilities by one another, but I guess that makes since as we are using a randomized algorithm to choose x,y,z.
Refangs
I think 3D (or any multidimensional random variable) can be thought of as a vector of random variables (I was initially confused about this). For the uniform 3D random variable on the slide X_i can be thought of as a vector [X Y Z]^T where X, Y, and Z are random variables, and the PDF p(x,y,z) captures the likelihood that X = x, Y = y, and Z = z. In the uniform case X, Y, and Z must be independent (since a uniform distribution's PDF is just going to be a constant).
Interesting, it seems like x,y,z are independent samples which is why we can multiply their 3 respective uniform probabilities by one another, but I guess that makes since as we are using a randomized algorithm to choose x,y,z.
I think 3D (or any multidimensional random variable) can be thought of as a vector of random variables (I was initially confused about this). For the uniform 3D random variable on the slide X_i can be thought of as a vector [X Y Z]^T where X, Y, and Z are random variables, and the PDF p(x,y,z) captures the likelihood that X = x, Y = y, and Z = z. In the uniform case X, Y, and Z must be independent (since a uniform distribution's PDF is just going to be a constant).