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Explain what separates a probability density function from a probability mass function.

FRM® Part I · Quantitative Analysis · Random Variables · FRM1.QA.2.d

The idea

A probability mass function belongs to a discrete variable and assigns a genuine probability to each possible value, so its values lie between 0 and 1 and add up to 1. A probability density function f(x) belongs to a continuous variable and does not give probabilities at points. The probability that a continuous variable equals any single value is zero. Instead, probability is the area under the density curve: P(a < X < b) is the integral of f(x) from a to b, and the total area is 1. Because a density is a height rather than a probability, it can exceed 1. On a uniform distribution over a range of width 0.25, for example, the height is: Density = 1 ÷ 0.25 = 4. The cumulative distribution function F(x) is the integral of the density up to x, and the density is its derivative. The usual mistake is reading a density value as a probability.

A practice question

From the question bank, with the reason each wrong option is wrong.

X is a continuous random variable with probability density function f(x). What is P(X = 0.02) exactly?

  1. f(0.02), the height of the density at that point.Why not: Treats the density value as a probability; it is only a height.
  2. F(0.02), the cumulative probability up to that point.Why not: Confuses the probability at a point with the CDF, which covers everything up to it.
  3. A positive number below 1 that depends on how steep f is at 0.02.Why not: Applies discrete-variable intuition; a continuous variable has no probability mass at any single value.
  4. Zero, because probability is an area under the density and a single point has no width. - correct

For a continuous variable, probability comes from integrating the density over an interval, so any single point carries probability zero.

A harder one

A loss ratio X is uniformly distributed on the interval from 0 to 0.25. What is P(0.05 < X < 0.15)?

  1. 4.00Why not: Reads the density value 1/0.25 = 4 as the probability.
  2. 0.10Why not: Uses the interval width 0.15 - 0.05 alone, without multiplying by the density of 4.
  3. 0.40 - correct
  4. 0.60Why not: Gives the probability outside the interval, 1 - 0.40, instead of inside.

The density is 1 / 0.25 = 4. Probability = area = density x width = 4 x 0.10 = 0.40.

The hardest tier

A random variable on [0, 2] has density f(x) = c x² for a constant c. What is P(X > 1)?

  1. 0.125Why not: Computes P(X < 1) instead of P(X > 1).
  2. 0.778Why not: Normalises over [0, 1] (c = 1/3) rather than [0, 2], so the density does not integrate to 1.
  3. 0.500Why not: Assumes a uniform distribution, so half the range carries half the probability.
  4. 0.875 - correct

The integral of c x² over [0, 2] is 8c/3 = 1, so c = 3/8. Then P(X < 1) = (3/8)(1/3) = 0.125 and P(X > 1) = 1 - 0.125 = 0.875.

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