Question

A square dartboard has side length 1, and is oriented as a diamond shape.

If I throw a dart uniformly at this dartboard, let <X,Y> be the x- and y- coordinates of the point the dart hits, with the center of the dartboard at the origin.

(a) Write the probability density function f_x(a) describing the distribution of X.

(b) compute EX (expected value) and Var[X] (Variance)

(c) Are X and Y independent? Why or Why not?

Answer 1

The given PDF is uniform in the square region which has an area 2. Thus,

The support is sketched below.

a) The marginal PDF of is

b) The expected value of is

Now,

The variance is

c) By symmetry,

Since , X and Y are not independent.