If X and Y are independent random variables each uniformly distributed on [0, 1], what is the probability P(X + Y ≤ 1)?

Answer and explanation

Correct answer: D. 1/2

  1. Step 1: The joint sample space is the unit square [0, 1] x [0, 1] with area 1.
  2. Step 2: The event X + Y ≤ 1 forms a right triangle with vertices (0, 0), (1, 0), and (0, 1).
  3. Step 3: Area of this triangle = (1/2) base height = (1/2) 1 1 = 1/2.
  4. Step 4: Since the joint pdf is uniform (f(x, y) = 1), P(X + Y ≤ 1) = 1/2.

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