1 Darts revisited Consider a circular dart board with radius one.

Q. 1 Darts revisited

Consider a circular dart board with radius one. I throw a dart at the board and record

the (X,Y) position (note that the dart will land in the board, and (X,Y) are uniformly distributed on the

board).

1. Find the marginal distribution of X.

2. Let R2 = X2 +Y2

(i.e. R is the distance from the center)

Find E[R] and E[R2]. (Please note that E[X]2 6=E[X2]

unless X is a constant random variable).

3. Let B∼Uniform[0,1]

and independent of R. Calculate

E[(R−B)2]