Description
1. Let X and Y be i.i.d. N (0, 1), and let S be a random sign (1 or −1, with equal
probabilities) independent of (X, Y ).
(a) Determine whether or not (X, Y, X + Y ) is MVN.
(b) Determine whether or not (X, Y, SX + SY ) is MVN.
(c) Determine whether or not (SX, SY ) is MVN.
2. Let X and Y be i.i.d. N (0, 1) r.v.s, T = X +Y , and W = X −Y . Show that T and W
are independent using two methods: 1) properties of MVN and 2) change of variables.
3. Let (X, Y ) denote a random point in the plane, and assume that the rectangular
coordinates X and Y are i.i.d. N (0, 1) r.v.s. Find the joint distribution of R and Θ
(shown in the following figure). Are R and Θ independent?
4. (a) Let X and Y be i.i.d. Expo(λ), and transform them to T = X + Y , W = X/Y .
Find the marginal PDFs of T and W, and the joint PDF of T and W.
(b) Let X, Y, Z be i.i.d. Unif(0, 1), and W = X + Y + Z. Find the PDF of W using
convolution.
(c) Let X and Y be i.i.d. Expo(λ) r.v.s and M = max(X, Y ). Show that M has the
same distribution as X + 1
2Y using two methods: 1) properties of the Exponential
and 2) convolution.
5. Programming Assignment:
(a) Use the Box-Muller Method to obtain the samples from the standard normal distribution N (0, 1). You need to plot the pictures of both histogram and the theoretical
PDF.
(b) Based on (a), generate samples from the standard bivariate Normal distribution,
where the correlation is ρ ∈ (−1, 1), and the marginal PDFs are both N (0, 1).
(c) According to the following picture format, plot the joint PDFs and the corresponding contours of standard bivariate Normal distribution with correlation ρ =
0, 0.3, 0.5, 0.7, 0.9.
⇢ = 0.8 ⇢ = 0.4 ⇢ = 0 ⇢ = 0.4 ⇢ = 0.8
fX,Y (x, y)
x
y
(a) (b) (c) (d) (e)
fX,Y (x, y) fX,Y (x, y) fX,Y (x, y) fX,Y (x, y)
x x x x
y y y y