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IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables

X and Y have joint density f(x,y) = 2 for 0 < x < y < 1 (zero elsewhere). What is the marginal density of X for 0 < x < 1?

The marginal density of X is 2(1 − x) for 0 < x < 1. For fixed x the variable y runs from x up to 1, so integrating the constant density 2 over that interval gives 2(1 − x).

  1. A2x
  2. B2(1 − x)Correct
  3. C2y
  4. D1
  5. 2(1 − x²)

Explanation

For a given x, y ranges from x to 1, so f_X(x)=∫ from x to 1 of 2 dy = 2(1−x). The option 2x would come from integrating y over 0 to x, which is the marginal of Y.

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