# Question

Suppose a point in three- dimensional Cartesian space, (X, Y, Z) is equally likely to fall anywhere on the surface of the hemisphere defined by X2 + Y2 + Z2 = 1 and Z ≥ 0.

(a) Find the PDF of Z, fZ (z).

(b) Find the joint PDF of X and Y, f X, Y (x, y).

(a) Find the PDF of Z, fZ (z).

(b) Find the joint PDF of X and Y, f X, Y (x, y).

## Answer to relevant Questions

Let X, Y and Z be a set of independent, zero- mean, unit- variance, Gaussian random variables. Form a new set of random variables according to U = X V = X + Y W = X + Y +Z. (a) Find the three one- dimensional marginal PDFs, ...Let represent a three- dimensional vector of random variables that is uniformly distributed over the unit sphere. That is, (a) Find the constant c. (b) Find the marginal PDF for a subset of two of the three random ...Consider the random sequence Xn = X / (1 + n2), where is a Cauchy random variable with PDF, Determine which forms of convergence apply to this random sequence. Prove that if a sequence converges in distribution to a constant value, then it also converges in probability. This one together constitute an alternative proof to the weak law of large numbers. Suppose we wish to estimate the probability, PA, of some event A as outlined .As motivated by the result, suppose we repeat our experiment for a random number of trials, N. In particular, we run the experiment until we ...Post your question

0