Someone asked on piazza if the dx dy dz were in the wrong order. Does it make a difference?
theandrewchan
Oh yeah student answer says that the differential is over volume anyway, which is dV=dx*dy*dz=dz*dy*dx. But dz dy dx would be more clear.
VioIchigo
@theandrewchan wait, but this is only true when you also change the order of the x0 x1, y0 y1, z0 z1 right?
jsc723
So do we always multiply the sample average with the volume of the integral to get F_N for the uniform estimator?
nathanpetreaca
For the case of a uniform estimator, yes I think it generalizes for all dimensions.
dtseng
@violchigo The order dx dy dz doesn't matter since the integral limits are independent of the other variables. If it's independent then it doesn't really matter what order you integrate it. However, if the integral limits depend on the other variables, then in general you do have to do some change of variables in x0x1 y0y1 z0z1.
AronisGod
An integral is a sum of the function over time. If the function is constant, it is that constant value times the dimensions of the "space" you are integrating over. The N samples aims at finding the "mean" of the function, s.t. we can estimate it as a constant mean, then multiply it by the dimensions of our integration to approximate the true integral.
Someone asked on piazza if the dx dy dz were in the wrong order. Does it make a difference?
Oh yeah student answer says that the differential is over volume anyway, which is dV=dx*dy*dz=dz*dy*dx. But dz dy dx would be more clear.
@theandrewchan wait, but this is only true when you also change the order of the x0 x1, y0 y1, z0 z1 right?
So do we always multiply the sample average with the volume of the integral to get F_N for the uniform estimator?
For the case of a uniform estimator, yes I think it generalizes for all dimensions.
@violchigo The order dx dy dz doesn't matter since the integral limits are independent of the other variables. If it's independent then it doesn't really matter what order you integrate it. However, if the integral limits depend on the other variables, then in general you do have to do some change of variables in x0x1 y0y1 z0z1.
An integral is a sum of the function over time. If the function is constant, it is that constant value times the dimensions of the "space" you are integrating over. The N samples aims at finding the "mean" of the function, s.t. we can estimate it as a constant mean, then multiply it by the dimensions of our integration to approximate the true integral.