How do you read the median and the p90 latency off a theoretical CDF F(t)?

## What the CDF is For any random variable X — discrete, continuous or a mixture — the cumulative distribution function is ``` F(t) = P(X <= t) ``` Unlike a PMF or a PDF, it always exists and always means the same thing: the probability accumulated at or below t. Its required properties follow directly: - **Non-decreasing.** For `a < b`, `F(b) - F(a) = P(a < X <= b) >= 0`, so the curve can never go down. - **Limits 0 and 1.** `F(t) -> 0` as t goes to negative infinity and `F(t) -> 1` as t goes to positive infinity. - **Right-continuous.** This is a consequence of defining F with `<=` rather than `<`; at a jump, the value at the jump point is the upper one. - **Interval probabilities by subtraction.** `P(a < X <= b) = F(b) - F(a)` — the single most useful line in practice. ## Reading quantiles: invert, do not read heights A quantile is a value on the x-axis, so you enter the plot from the y-axis. Formally the quantile function is ``` Q(p) = the smallest t such that F(t) >= p ``` - **Median** = `Q(0.5)`: the smallest t with `F(t) >= 0.5`. Half the probability sits at or below it. - **p90** = `Q(0.9)`: the smallest t with `F(t) >= 0.9`. Nine tenths of requests are at or faster than it. Graphically: find 0.9 on the vertical axis, travel horizontally until you meet the curve, then drop straight down to the time axis and read the latency. The classic error is doing it the other way round — reading `F(t)` at some t and calling that a percentile. `F(t)` is a probability; a percentile is a value of t. A second classic error is direction: p90 is where `F(t)` first reaches **0.9**, not 0.1. The 0.1 level gives the p10, the fast end. ## The survival function Many operational questions are naturally phrased about the tail, and the survival function is the same information mirrored: ``` S(t) = 1 - F(t) = P(X > t) ``` It starts at 1, is non-increasing, and tends to 0. Uses: - "What share of requests breach a 500 ms threshold?" is `S(500)`. - "What is the p90?" is equivalently "the smallest t where `S(t)` first falls to 0.1". - "What share of requests land between 200 ms and 500 ms?" is `S(200) - S(500)`, which equals `F(500) - F(200)`. When a distribution is heavy-tailed, the survival curve is the more legible of the two, because the interesting behaviour is where F is crawling from 0.99 to 1 and visually flat. ## The two edge cases interviewers probe **Flat stretches.** If F is constant across an interval, no probability lives there. If it happens to be flat at exactly the level 0.5, then every t across that whole flat run satisfies `F(t) >= 0.5` at its right end and the median is not unique; the "smallest t" convention picks a single representative. This is why the definition uses an inequality and a smallest-value rule rather than solving `F(t) = 0.5`. **Jumps.** Discrete or mixed variables have jumps in F, and then a level like 0.9 may be skipped over entirely: F leaps from 0.85 to 0.93 at some t, and no value satisfies `F(t) = 0.9`. The `F(t) >= p` formulation still returns that jump point, which is why it is stated as an inequality. It is also why quantiles of discrete variables can look coarse — several probability levels map to the same value. ## Why the CDF is the object worth reasoning in Densities are shape-friendly but their heights are not probabilities; PMFs only exist for discrete variables. The CDF gives you probabilities directly, in `[0, 1]`, for any variable, and it supports the operations you actually want: threshold probabilities by evaluation, interval probabilities by subtraction, quantiles by inversion, tails through `S(t)`. ## How to say it in an interview "F(t) is `P(X <= t)`, so I invert it: the median is the smallest t with `F(t) >= 0.5`, the p90 the smallest t with `F(t) >= 0.9`. On a plot, enter at 0.9 on the y-axis, go across to the curve, drop to the x-axis. For SLO questions I flip to `S(t) = 1 - F(t)`, so the breach rate at 500 ms is `S(500)`. Watch for flat stretches, where the quantile is non-unique, and jumps, where a level is never hit exactly."