Sampling Theorem
State the 1-dimensional sampling theorem and sketch a proof. Explain the notion of conditional density function in relation to joint and marginal densities for dependent variables. For a 3-dimensional vector random variable, whose components are statistically dependent, derive a mathematical sampling algorithm as an extension of the 1-dimensional case.
Based on your algorithm, write a simulation in MATLAB to generate i.i.d. samples for the following joint PDF in 3-dimensions:
fxyz(x, y, z) = k(x + y + 2)
supported on the unit cube
Sx, y, z < 1