PMF, PDF and CDF

## The short answer, then the reason Under a continuous model the answer is exactly 0, and the reason is structural rather than a matter of "very unlikely". A continuous random variable X is described by a density f, and the defining property of a density is that probability equals **area**: ``` P(a <= X <= b) = area under f between a and b ``` Now shrink the interval onto a single point. Set a = b = 200. The region under the curve has width 0 and finite height, so its area is 0. There is no probability left to assign. This holds for every point, including the one you eventually observe. ## The same statement in CDF language The cumulative distribution function is `F(t) = P(X <= t)`. Any point mass shows up as a **jump** in F: the probability of the single value 200 is the size of the jump of F at 200, `F(200) - F(200^-)`, where `F(200^-)` is the limit from the left. A continuous variable has a continuous CDF — no jumps — so every such difference is 0. This is also the cleanest way to see the contrast with a discrete variable. If X counts retries and `P(X = 3) = 0.12`, the CDF of X steps up by 0.12 at 3, and that step *is* the PMF value. A PMF assigns positive mass to individual points; a PDF never does. ## Probability zero is not impossibility Candidates often flinch here, because "probability 0" sounds like "cannot happen". The precise statements are: - The **impossible** event is the empty set: no outcome belongs to it. - A **probability-zero** event can be non-empty and can occur. Any single latency value has probability zero, yet the request does return some latency. The model says only that no individual value is singled out with positive mass; the mass lives on intervals. Analogously, picking a point at random on a line segment lands somewhere, but every particular point had probability zero of being the one. ## What to ask instead The question becomes well-posed the moment you name an interval or a threshold: - `P(199.5 < X <= 200.5) = F(200.5) - F(199.5)` — a band around 200. - `P(X > 500) = 1 - F(500)` — the tail beyond an SLO threshold. - `P(a < X <= b) = F(b) - F(a)` — the general form. The density value `f(200)` is *not* one of these. It is probability per unit of time, not a probability. It is useful for comparing relative plausibility near different values, and `f(200) * w` approximates `P(200 - w/2 < X <= 200 + w/2)` for a small width w, but on its own it answers nothing about the event "exactly 200". ## Open versus closed endpoints Because each point contributes 0, all four of these coincide for a continuous X: ``` P(a < X < b) = P(a <= X < b) = P(a < X <= b) = P(a <= X <= b) ``` For a discrete variable they differ by the PMF values at the endpoints, which is exactly why interview problems on discrete variables are fussy about `<` versus `<=` and continuous ones are not. ## The honest caveat about real data Real timers report rounded values — say whole milliseconds — so the recorded quantity is genuinely discrete and "exactly 200" has positive probability at the level of the *recording*. The continuous density is a model of the underlying quantity, chosen because it is analytically convenient and because the rounding grid is fine relative to the spread. A strong answer says both things: zero under the continuous model, and if you truly need the probability of the recorded value 200, you are asking about the interval the rounding maps onto, `P(199.5 < X <= 200.5)`. ## How to say it in an interview "Zero — a point has no width, so no area. Probability zero is not impossibility; the request still returns a value. If you want a number, give me an interval: `F(200.5) - F(199.5)`." Then, if there is time, add the CDF-jump framing and the discrete contrast.