A point is randomly selected with uniform probability in the X-Y plane within…
2004
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
Show answer & explanation
Correct answer: D
Let the randomly selected point be (x, y). Since p is the length of its position vector, p^2 = x^2 + y^2. Here x is uniform on [0, 1], so E[x^2] = integral from 0 to 1 of x^2 dx = 1/3. Also y is uniform on [0, 2], so E[y^2] = (1/2) integral from 0 to 2 of y^2 dy = 4/3. Therefore E[p^2] = E[x^2] + E[y^2] = 1/3 + 4/3 = 5/3. Hence option D is correct.