< Probability and Statistics Definitions < Cumulative Mass Function
What is the Cumulative Mass Function?
The Cumulative Mass Function (CMF) gives the probability a random variable will take on a value less than or equal to a specific value.
While the term “Cumulative Distribution Function (CDF)” is more universal and applies to both discrete and continuous random variables, the CMF only applies to discrete random variables. The CMF and the CDF are functionally equivalent in the context of discrete random variables: both the CMF and CDF will provide cumulative probabilities up to a certain point x. The distinction lies primarily in terminology and the emphasis on the discrete nature of the random variable when referring to the CMF.
Formal Definition of Cumulative Mass Function
The CMF is formally defined as follows:
Where
- X = a random variable
- x = the specific value you want to calculate the probability for (i.e. X ≤ x)
- P(X = t) = the probability mass function (PMF) of X, which gives the probability of X = t.
CMF Properties
- The CMF is a non-decreasing function of x — this is perhaps not surprising as cumulative probabilities can only grow or remain constant.
- The CMF is a right-continuous function, which means that for any point x, the cumulative value CMF(x) will equal its value from the right. In other words, limδ→0(x + δ) = CMF(x).
- The probability of the random variable X taking a value less than negative infinity is zero. That’s: limx→-∞CMF(x) = 0.
- The probability equals 1 for the entire range: limx→∞CMF(x) = 1.
- The cumulative mass function is a step function, which increases in step at points where the random variable’s probability mass is positive.
CMF vs PMF
The PMF gives us the cumulative probability up to point x, while the PMF gives us the probability X will take on an exact value x. This means that the CMF is useful for calculating probabilities over intervals, the PMF is useful for calculating individual outcomes.