A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0,0), (1,0)...

Question

A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0,0), (1,0), (1,2) and (0,2). If p is the length of the position vector of the point, the expected value of p^2 is

A. 2/3
B. 1
C. 4/3
D. 5/3

Accepted Answer

E[p^2] = 5/3. Answer: D. A point (x, y) is chosen uniformly at random from the rectangle [0,1] x [0,2]. The position vector has length p = sqrt(x^2 + y^2), so p^2 = x^2 + y^2. We need E[p^2] = E[x^2 + y^2] = E[x^2] + E[y^2]. Since x and y are independent uniform random variables, we compute their second moments separately.

A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0,0), (1,0), (1,2) and (0,2). If p is the length of the position vector of the point, the expected value of p^2 is A. 2/3 B. 1 C. 4/3 D. 5/3