Consider two points that are independently placed on a line of length 10, at locations X and Y. Thus the joint density of X and Y is
fX,Y(x,y) = 1/100
for 0 ≤ x ≤ 10 and 0 ≤ y ≤ 10,
and fX,Y(x,y) = 0 otherwise.
a. First, try to show that, if Z denotes the distance between X and Y (so 0 < Z < 10), then the cumulative distribution function FZ(z) of Z is
FZ(z) = 1/5z - 1/100 z2
for 0 ≤ z ≤ 10
and FZ(z) = 0 for z ≤ 0 and FZ(z) = 1 for z ≥ 10.
b. Regardless of whether you can accomplish the part above, just find fZ(z), by differentiating FZ(z).
c. Use the density fZ(z) to find E(Z), i.e., the expected distance between X and Y.