7. An archer shoots an arrow at a circular target of radius 1 metre in such a way that

the probability that the arrow hits a region A on the target is proportional to the

area of the region A (i.e. the arrow is equally likely to hit any point on the target).

In addition, suppose that the archer never misses the target.

(a) Determine the probability that the arrow lands closer to the centre of the cir-

cular target than to the edge of the target.

(b) Let X denote the distance, in centimetres, between where the arrow hits the

target and the centre of the target. Determine Im X, the image of X, and find

the distribution function of X and sketch it.

Determine: (i) P(X = 20), and (ii) P(30 < X < 90).

(c) Now suppose that the archer scores points as follows: 50 points if X 10cm,

25 points if 10cm < X 40cm, 10 points if 40cm < X 70cm, and 5 points

otherwise. Let Y denote the archer's score.

Determine Im Y , the image of Y , and find the distribution function of Y