discrete vs continuous distributions
Some quantities come in separate, countable lumps; others can take any value along a smooth scale. The number of claims this year is 0, 1, 2, or 3 — never 2.7. But the size of a claim can be $2,719.43 or any amount in between. The first kind of random variable is called discrete; the second is called continuous. Which type you have decides how you describe its probabilities.
A discrete random variable has a list (possibly infinite) of separate possible values, and we attach a genuine probability to each one — the probability mass at that value. A continuous random variable can take any value in a range, and here a strange thing happens: the probability of any single exact value is zero, because there are infinitely many values. Instead we describe a continuous variable by a density curve, and probability is the area under that curve over an interval. So for discrete variables we sum probabilities; for continuous ones we integrate (find areas).
Actuaries deal with both constantly, and often together. Claim frequency (how many claims) is naturally discrete — Poisson, binomial, negative binomial. Claim severity (how large) is naturally continuous — lognormal, gamma, Pareto. The total loss combines a discrete count with continuous sizes, which is why aggregate loss models splice the two. Picking the wrong type — treating a count as continuous or a smooth loss as a handful of buckets — quietly distorts the answer.
Number of hospital admissions next year: discrete (0, 1, 2, ...). Cost of each admission: continuous (any positive dollar amount). A health model uses a count distribution for the first and a continuous one for the second.
Counts are discrete; amounts are usually continuous — and real models often need both.
For a continuous variable the probability of any exact single value is zero; only intervals carry positive probability, so 'P(loss = exactly $1000)' is meaningless while 'P(loss between $900 and $1100)' is fine.