Mathematics question
1. It is common for engineers to work with the “error function”
erf(z) = 2
√
π
Z z
0
exp
−x
2
dx
instead of the standard normal probability distribution function Φ, which we defined as:
Φ(z) = Z z
−∞
1
√
2π
exp
−
x
2
2
dx.
Show that the following relationship between Φ and the function erf holds for all z:
Φ(z) = 1
2
+
1
2
erf
z
√
2
.
2. Suppose that a pair of random variables (X,Y) is uniformly distributed on the vertices of the square [−1, 1] × [−1, 1]: i.e., the joint mass function has nonzero probability on (−1, −1), (1, −1), (−1, 1), and (1, 1) with each of these four points occurring
with probability pX,Y(x, y) = 1
4
.
(a) Compute P(X
2 + Y
2 < 1)
(b) Compute P(2X − Y > 0)
(c) Compute P(|X − Y| < 2)
DETAILED ASSIGNMENT
