A discrete distribution describes uncertainty for variables that take countable outcomes, such as defaults, claims, births, or number of purchases.
Core Mechanics
If (X) is discrete, its probability mass function (pmf) is:
[ p(x) = P(X=x) ]
with:
[ \sum_x p(x) = 1 ]
Expected value is computed by weighted summation:
[ E[X] = \sum_x x,p(x) ]
Economic Applications
Discrete distributions are standard in count-data and event-data settings, including labor transitions, firm entry/exit counts, and credit default events.
Practical Modeling Context
Choosing an inappropriate distribution can bias standard errors and marginal effects. Analysts often test whether variance behavior matches the assumed pmf family before final estimation.