3.24 Coverage of a square lattice by a rectangle A city's geographical structure is being placed on a computer. All coordinate positions are being' quantized, where the unit of quantization is 500 feet. The quantization points comprise a lattice that runs east-west and north-south. The board of elections wishes to know how many quantization points will be contained in an arbitrary rectangular election district of dimension t (east-west) and m (north-south).
    Assume that the location of the election district on the lattice can be modeled as random (but the sides are parallel to the two directions of the lattice). Let N be the number of lattice points contained within the election district.

a. Show that

E[N] = ell.gif (55 bytes)m

b. Let ell.gif (55 bytes) = p + q, m = P + Q( 0 leq.gif (53 bytes) q, Q < 1). Show that

pg173a.gif (3263 bytes)