How do you describe a customer spend variable with a point mass at 0 and continuous spend above it?

## Not every variable is discrete or continuous Introductory treatments present two kinds of random variable — discrete with a PMF, continuous with a PDF — and a lot of real business metrics are neither. Spend per customer over a window is the canonical case: most customers buy nothing, so a large chunk of probability sits exactly on the value 0, while the buyers' spend varies smoothly over positive amounts. Such a variable is called **mixed**: its distribution has a discrete atom glued to a continuous part. ## Why neither single object works - **A PMF fails.** It assigns positive mass to individual values and sums to 1. The positive-spend region cannot be described that way, since each individual spend amount carries zero probability there. - **A PDF fails.** A density's probability is area, and the area over the single point 0 is zero, no matter how the density is defined. So no density can carry the non-buyer mass p. There is no way to "put a tall spike at 0" that fixes this within ordinary densities. ## The CDF is the object that always works `F(x) = P(X <= x)` is defined for every random variable, and it represents both parts naturally. Let p be the probability of spending nothing. Then - `F(x) = 0` for `x < 0`, - `F(0) = p` — a **jump** of height p at 0, and the jump height is exactly the point mass, - `F(x)` rises continuously for `x > 0`, approaching 1. Atoms are jumps; continuous parts are smooth rises. Reading a distribution's CDF tells you immediately which regions carry mass on points and which spread it out. ## The mixture representation The most usable description is a two-part decomposition. Let B be the indicator of buying, `P(B = 0) = p`, and let G be the CDF of spend conditional on buying, with `G(0) = 0`. Then for `x >= 0` ``` F(x) = p + (1 - p) * G(x) ``` Read it as: with probability p you land on the atom at 0; otherwise you draw from the buyers' continuous distribution. This mirrors how such data is usually modelled — a purchase probability and a conditional spend distribution, estimated as two separate pieces — and it keeps the two mechanisms (whether someone buys, and how much they spend if they do) from being blurred into one shape. ## What this does to quantiles This is where the mixed structure bites in reporting. Suppose `p = 0.7`. The quantile at level q is the smallest x with `F(x) >= q`. Since `F(0) = 0.7`: - Every level `q <= 0.7` is already satisfied at `x = 0`, so **all** those quantiles equal 0. The median spend is 0. - The overall p90 solves `0.7 + 0.3 * G(x) = 0.9`, so `G(x) = 2/3`: the overall 90th percentile equals the buyers' 66.7th percentile. Two lessons follow. First, a median of 0 is not a broken metric — it is the correct answer, and it is uninformative precisely because the atom dominates. Second, overall percentiles above the atom are silently percentiles of the buyer distribution at a shifted level, so comparing overall p90 across two periods conflates a change in the buy rate with a change in the amount spent. ## How to report it The honest summary is two numbers rather than one: the share of customers with zero spend, and the distribution of spend among buyers. That separates the two mechanisms, and it makes a movement interpretable — either more people bought, or buyers spent differently, or both. Collapsing to a single overall percentile hides which one moved. What you should not do is quietly drop the zeros and report only buyers, while labelling the result as spend per customer. That is a different quantity — conditional on purchase — and it will systematically overstate the per-customer figure. ## The generalisation Any distribution decomposes into a discrete part and a continuous part, and mixed variables appear well beyond spend: any metric with a floor or a cap collects an atom there, and any quantity that is exactly zero unless an event occurs has an atom at zero. The recognition skill is the valuable one: see a spike at a boundary value, ask whether it is a genuine point mass rather than a narrow peak, and if it is, switch to the CDF or mixture description rather than forcing a density onto it. ## How to say it in an interview "It is a mixed distribution, so no single PMF or PDF describes it. I would use the CDF, which jumps by the non-buyer share at 0 and rises smoothly above, or equivalently a mixture: buy probability times a conditional spend distribution. Practically, if 70% spend nothing, the median is 0 and the overall p90 is the buyers' two-thirds quantile — so I would report the zero share and the buyer distribution separately."