Problem 2

p_xy (x, y) = 1/(8π) ^(-(x^2 + y^2 + 6x - 2y + 10)/8)

a) What type of distribution is p_xy(x, y)?

p_xy (x, y) was supposed to be a Gaussian distribution, but the coefficient in the quiz was 1/ ... ent was positive, which makes it integrate to a value not equal to 1, therefore it was not a pdf .

b) Are x and y independent?

Yes, they are independent. This can be shown by showing that p(x)p(y) = p(x,y).

1/(8π) ^(-(x^2 + y^2 + 6x - 2y + 10)/8) = (1/(8π)^(1/2) ^(-(x^2 + 6x + 9)/8)) (1/(8π)^(1/2) ^(-(y^2 - 2y + 1)/8))

c) Are x and y correlated?

No, x and y are not correllated. If random variables are independent, they are uncorrelated.

d) What are the mean and variance values of x?

By inspection of part (b)

μ_x = -3 σ_x^2 = 4

Created by Mathematica  (October 8, 2003)