solution

(3) We say a random variable z ∼ kχ2n follows a kχ2 n distribution if z/k ∼ χ2
n. Given an array gij ∼M (μij,σ2) of random variables, representing a noisy sampled function g(x,y), the partial derivatives can be derived from
gx(i,j) = (g(i+ 1,j) −g(i−1,j))/2 , gy(i,j) = (g(i,j + 1) −g(i,j −1))/2 . (2.232)
Give the standard deviations of the two partial derivatives and their covariance. What is the distribution of the squared magnitude m2(i,j) := |∇g(i,j)|2 = g2
x(i,j) + g2 (i,j) of the gradient ∇g = [gx,gy]T? Hint: Which distribution would m2 follow if the two random variables gx and gy were standard normally distributed?