## 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