Description

3.
pR,Θ (r,θ) = pX1,X2 (x1, x2)J
4. The marginal distribution (1.3.1)
Z 2π pR(r) = pR,Θ (r,θ) dθ
0
5. The Laplace transform of pY(y) is given by (1.4.1)

MY(s) = E e−sY
6. The unit step function is defined as (1.5.1)
 u(y) = 1 y ≥ 0
0 otherwise
2 Problems
1. Let X1, X2 ∈ N ∼ (0,1) be i.i.d. Find pX1,X2 (x1, x2).
2. Let (1.6.1)
X1 = RcosΘ (2.2.1)
X2 = RsinΘ
Find pR,θ (r,θ).
3. Find pR(r).
4. Find pΘ(θ). (2.2.2)
1 Definitions
1. The pdf of X ∼Nµ,σ2 is defined as
fX (1.1.1)
2. The Jacobian matrix transforming R,Θ to X1, X2 is defined as
!
J(1.2.1)
2
5. Find the distribution of
Y = X12 + X22 (2.5.1)
6. Find the Laplace transform of e−yu(y) 7. Find the Laplace transform of pX12(x1).
8. Find the Laplace transform of pY(y) using (2.5.1).
9. Find the distribution of Y.