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FRM Part I · FRM Exam Part I · Random Variables

Which statement about probability mass and density functions is correct?

For a continuous variable, the probability at any single point is zero because probability equals area under the density. The density height itself can exceed 1, such as 2 for a uniform on 0 to 0.5, as long as total area is 1.

  1. AA probability density function value f(x) at a point is the probability that X equals x, and cannot exceed 1.
  2. BFor a continuous random variable, P(X = x) is zero for any single x, and the density f(x) can exceed 1 over a narrow range.Correct
  3. CThe CDF of a discrete random variable is continuous because it accumulates probabilities from the PMF.
  4. DThe area under a PMF is found by integrating it across the support.

Explanation

For continuous variables probability is area, so any single point has probability zero, yet the density height can exceed 1 (for example, uniform on 0 to 0.5 has density 2). A density value is not a probability. A discrete CDF is a step function, and PMF probabilities are summed rather than integrated.

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