Aug 04

Probability functions: PMF vs. PDF vs. CDF (Quant: Stat)

by David Harper, CFA, FRM, CIPM

FRM |

Learning objective: Define probability mass function, probability function, probability density function, cumulative density function

If the random variable is discrete, then the probability mass function (PMF) is the probability that the discrete random variable assumes the exact value of x; i.e., PMF: P(X = x).

If the random variable is continuous, then the probability density function (PDF) is the probability that the continuous random variable assumes a value over an interval or range; i.e., PDF: P(x1 £ X £ x2).

The cumulative density function (CDF) associates with either a PMF or PDF (i.e., the CDF can apply to either a continuous or random variable). The CDF gives the probability the random variable will be less than, or equal to, some value. CDF: P(X £ x).

  Density Cumulative
Discrete PMF, P (X = x) CDF (step function)
Continuous PDF, P (X <= x) CDF (continuous curve)

In summary:

  • Probability mass function (PMF): probability that discrete random variable will exactly equal a discrete value
  • Probability density function (PDF): probability that continuous random variable will fall within an interval
  • Cumulative density function (CDF): probability that either a discrete or continuous random variable with take a value of less than, or equal to, a certain value.

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