🔔 Gaussian and Exponential Models

By Better Wise

The last lesson handled columns that count things: items_count fit a Poisson, a yes/no flag fit a Bernoulli. This lesson covers the other two column types you meet in almost every dataset question: a measurement (a price, a weight, a temperature) and a waiting time (minutes to delivery, seconds to the next click). The Gaussian answers percentile questions about measurements. The exponential answers "how long until" questions about gaps. Both connect back to the Poisson: if events arrive as Poisson counts at rate $\lambda$, the gaps between them are exponential with that same $\lambda$.

By the end of this lesson you should be able to:

  • Fit and interpret a Gaussian from a mean and standard deviation
  • Turn "what share is above X?" or "what value marks the top 10%?" into a z-score lookup
  • Use the exponential CDF to answer waiting-time questions on a real column
  • Spot when a Gaussian is the wrong model for a time column
  • Explain memorylessness and why it matters for interpretation

We work with quickcart_orders.csv, which has order_value_usd (a measurement) and delivery_minutes (a waiting time).

A measurement column: order_value_usd

The Gaussian $N(\mu, \sigma^2)$ is a symmetric bell curve described by two numbers: the center $\mu$ and the spread $\sigma$.

$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) $$

Almost every question you actually get asked is a percentile question, and all of them go through the same move: standardize, then read the table.

$$ z = \frac{x - \mu}{\sigma} $$

The z-score says how many standard deviations the cutoff sits from the mean. You then look up the tail area. Useful anchors: $z = 0.6745$ is the 75th percentile, $1.2816$ is the 90th, $1.6449$ is the 95th. Inverting the formula gives the percentile back: $x = \mu + z\sigma$.

Worked example 1. Past orders give $\mu = 62$ and $\sigma = 14$ dollars. What share of orders is above $80?

$$ z = \frac{80 - 62}{14} = 1.29 $$

The upper tail beyond $z = 1.29$ is about $0.099$, so roughly 10% of orders exceed $80.

Worked example 2. What order value marks the top 10%?

$$ x = 62 + 1.2816 \times 14 = 79.94 $$

So about $80. The two answers agree, which is a good consistency check: the 90th percentile is the value with 10% above it.

Worked example 3. Share of orders between $48 and $76 is $z = -1$ to $z = +1$, which by the empirical rule is about 68%. Saying "roughly two thirds of orders fall within one sigma" is a sentence a stakeholder understands instantly.

A waiting-time column: delivery_minutes

Delivery time cannot be negative and it is right-skewed: many short deliveries, a long thin tail of slow ones. The exponential model fits this shape with a single parameter.

$$ f(t) = \lambda e^{-\lambda t}, \qquad F(t) = P(T \le t) = 1 - e^{-\lambda t}, \qquad t \ge 0 $$

Here $\lambda$ is a rate per unit time, not a time. The mean wait is $1/\lambda$. In delivery_minutes the mean is 28 minutes, so $\lambda = 1/28 \approx 0.0357$ per minute. This is the easiest place to make a mistake: if someone hands you a mean, invert it before touching the formula.

Worked example 4. Probability a delivery takes longer than 45 minutes:

$$ P(T > 45) = e^{-45/28} = e^{-1.607} \approx 0.20 $$

About 20% of deliveries are slower than 45 minutes.

Worked example 5. Median delivery time: $\ln(2)/\lambda = 28 \times 0.693 \approx 19.4$ minutes. Note the median is well below the mean, which is the signature of a right-skewed column. If your mean and median are that far apart, do not fit a Gaussian.

Worked example 6. Suppose a colleague fits a Gaussian with mean 28 and standard deviation 16 instead. That model gives $P(T > 45) \approx 0.14$ (too low, it thins the tail) and puts about 4% of the probability below zero minutes, which is impossible. That is the argument for choosing the model from the shape of the column, not from convenience.

Memorylessness, and why it changes your interpretation

The exponential has a property no Gaussian has:

$$ P(T > s + t \mid T > s) = P(T > t) $$

Given a delivery has already taken 20 minutes, the chance it takes 25 more is the same as the chance a fresh order takes more than 25 minutes: $e^{-25/28} \approx 0.41$. Surviving the wait so far tells you nothing extra. This is a strong assumption, often false in the real world (traffic builds up), so state it explicitly when you use it.

flowchart TD
    Q["Which column is the question about?"]
    Q --> M["📏<br/>Measurement, roughly symmetric"]
    Q --> W["⏱️<br/>Waiting time or gap"]
    M --> G["Gaussian<br/>mu and sigma"]
    W --> E["Exponential<br/>rate lambda, mean 1 over lambda"]
    G --> Z["Standardize: z equals x minus mu over sigma"]
    E --> C["Tail: e to the minus lambda t"]
Question on your desk Model Formula to reach for
Share of orders above $80 Gaussian upper tail beyond $z$
Value that marks the top 10% Gaussian $x = \mu + 1.2816\sigma$
Share of deliveries slower than 45 min Exponential $e^{-\lambda t}$
Median or 90th percentile wait Exponential $\ln(2)/\lambda$, $\ln(10)/\lambda$

Reading the tails correctly: traps that cost marks

  • Check the shape before fitting. Plot a histogram. A Gaussian assumption on a skewed column gets the tail wrong, and the tail is where the interesting questions live.
  • Rate is not a time. $\lambda = 1/28$ per minute, mean 28 minutes. Confusing the two gives answers off by a factor of 784.
  • The mean is not the median. For any exponential, median $= 0.69 \times$ mean. Report both when the column is skewed.
  • "Above" means the upper tail. For $P(X > x)$ with a Gaussian, subtract the table area from 1. This is the single most common arithmetic slip.
  • Standard deviations, not units. Always convert to $z$ before comparing cutoffs from two different columns.
  • State the memorylessness assumption out loud when you condition on elapsed wait, because it is a modeling claim, not an arithmetic step.

Two exercises

  1. A premium segment has $\mu = 45$, $\sigma = 12$ for order_value_usd. (a) What share is above $60? (b) What value marks the 75th percentile?
  2. support_wait_minutes is exponential with mean 12. (a) $P(T > 20)$? (b) The median? (c) $P(T > 32 \mid T > 20)$?

Show the answers

  1. (a) $z = (60-45)/12 = 1.25$, upper tail $\approx 0.106$, about 10.6%. (b) $45 + 0.6745 \times 12 = 53.09$, about $53.
  2. $\lambda = 1/12$. (a) $e^{-20/12} = e^{-1.667} \approx 0.189$, about 19%. (b) $12 \times 0.693 \approx 8.3$ minutes. (c) By memorylessness, $e^{-12/12} = e^{-1} \approx 0.368$, about 37%.

What to carry forward

  • A Gaussian question is a percentile question: standardize to $z$, read the tail, invert for the percentile value.
  • An exponential question is a waiting-time question: get $\lambda$ as a rate, then use $e^{-\lambda t}$ for tails and $\ln(2)/\lambda$ for the median.
  • Choose the model from the column's shape and support (non-negative, skewed, symmetric), not from convenience.
  • Association is not the goal this time: the next lesson, Bayes' Rule and Base Rates, asks how to revise a belief about a single row (for example, is this late delivery actually late?) when base rates and a noisy test disagree.

Front: What does the Gaussian model N(mu, sigma^2) describe? Back: A roughly symmetric measurement column, summarized by its mean mu and standard deviation sigma.

Front: How is a z-score computed? Back: z = (x - mu) / sigma, the number of standard deviations x sits from the mean.

Front: Which z-value marks the 90th percentile? Back: About 1.2816, so the 90th percentile is mu + 1.2816 sigma.

Front: How do you handle a 'greater than' Gaussian question? Back: Standardize the cutoff, look up the area, then subtract it from 1 to get the upper tail.

Front: What share of a Gaussian lies within one and two standard deviations? Back: About 68% within one sigma and about 95% within two.

Front: In the exponential model, what is lambda? Back: A rate per unit time, not a duration; the mean waiting time is 1/lambda.

Front: What is the exponential CDF? Back: P(T <= t) = 1 minus e to the power minus lambda t, so P(T > t) = e to the minus lambda t.

Front: How does the exponential median relate to its mean? Back: Median = ln(2)/lambda, about 0.69 times the mean.

Front: What does memorylessness of the exponential say? Back: P(T > s + t | T > s) = P(T > t), so surviving a wait does not change the remaining wait distribution.

Front: How are Poisson counts and exponential gaps related? Back: If events arrive as Poisson counts at rate lambda, the gaps between events are exponential with that same lambda.

Front: Why is a Gaussian a poor model for delivery_minutes? Back: Times are non-negative and right-skewed, so a Gaussian puts mass below zero and understates the long tail.

Front: What is the closed form for the 90th percentile of an exponential? Back: ln(10)/lambda, roughly 2.303 divided by the rate.

Front: Which summary should you report for a skewed waiting-time column? Back: The median (and the mean alongside it), since the mean is pulled up by the long tail.

Three facts worth locking in with cloze practice: a cutoff becomes comparable across columns once you convert it to a _____, which counts standard deviations from the mean. In the exponential model, $\lambda$ is a _____, so the mean wait equals _____. The exponential is _____, meaning the remaining wait has the same distribution no matter how long you have already waited.