Continuous Distributions

Wait times, salaries, temperatures, response latencies, heights — these don't come in whole-number counts. They're continuous: between any two possible values there's always another. That one fact forces a different way of thinking about probability than the discrete world of Discrete Distributions, and it trips up almost everyone the first time.

Here's the shift. For a count, "what's the probability of exactly 3 tickets?" is a sensible question with a real answer. For a continuous quantity, "what's the probability the wait is exactly 5.000000... minutes?" has the answer zero — there are infinitely many possible values, so no single one carries any probability on its own. Instead, probability for continuous variables lives in intervals, and it's computed as area under a curve. Internalize that and the rest of this page falls into place.

The big idea: probability is area under the PDF

A continuous distribution is described by a probability density function (PDF) — a smooth curve. The curve itself is not probability. Probability is the area under the curve between two values:

P(a < X < b) = area under the PDF from a to b

The total area under any PDF is exactly 1 (something must happen). A single point has zero width, so it has zero area — which is why P(X = exact value) = 0 for continuous variables. You always ask about ranges.

Discrete PMF vs continuous PDF

The contrast with the discrete world is worth pinning down, because the mental model is genuinely different.

Uniform: every value equally likely

The Uniform distribution spreads probability evenly over an interval [a, b] — a flat PDF. It's the simplest continuous distribution and the natural model when you have no reason to favor any value over another in a range: a random timestamp within an hour, a spawn point along a line, the raw output of a random-number generator.

In scipy the convention is loc = a (the left edge) and scale = b - a (the width), so Uniform(0, 10) is stats.uniform(loc=0, scale=10). This loc/scale convention runs through almost every scipy distribution — learn it once here.

Exponential: waiting time until the next event

The Exponential distribution models the time between independent events that happen at a constant rate — the continuous partner of the Poisson from Discrete Distributions. If support tickets arrive Poisson-style at some rate, the gap between consecutive tickets is Exponential. Real uses:

• Wait times: time until the next customer, request, or arrival.
• Time-to-failure: how long a component lasts before breaking.
• Inter-event gaps: seconds between clicks, days between incidents.

In scipy, parameterize by scale = mean. So a process with a mean wait of 5 minutes is stats.expon(scale=5). (There's also a loc shift, but you'll almost always leave it at 0.)

The exponential is right-skewed: short waits are most likely, but there's a long tail of occasional long waits. Its defining quirk is memorylessness — having already waited 10 minutes tells you nothing about how much longer you'll wait. That's a strong assumption, and a reason it sometimes doesn't fit (a machine that wears out gets more likely to fail over time, violating memorylessness).

Normal: the bell curve (a first look)

The Normal (Gaussian) distribution is the famous symmetric bell curve, defined by a mean μ (center) and standard deviation σ (spread): stats.norm(loc=mu, scale=sigma). It models quantities that pile up around a center with symmetric tails — measurement errors, aggregated noise, many natural and human measurements. It's so central that it gets its own page next, The Normal Distribution, where we cover the 68–95–99.7 rule and z-scores. Here we just meet it as another continuous distribution and practice the same area-based questions.

Quantiles: inverting the question with ppf

So far we've gone value → probability with cdf. Often you need the reverse: probability → value. "What wait time are 90% of waits below?" "What score puts you in the top 5%?" That's the percent-point function (ppf), the inverse of the CDF, also called the quantile function.

ppf(0.9) returns the value x such that P(X ≤ x) = 0.9 — the 90th percentile. It's the tool for setting thresholds, SLAs, and cutoffs.

Reading a density plot correctly

One more guard against the height-is-probability trap. When you plot a PDF, the y-axis is density, and you should read relative heights and areas, never absolute heights as probabilities.

Check your understanding

QuestionSelect one

For a continuous random variable X, what is P(X = 7.0) — the probability it equals exactly 7.0?

It equals the height of the PDF at 7.0

It is 0, because a single exact point has zero width and therefore zero area under the PDF

It is small but positive, roughly the PDF height times a tiny number

It cannot be computed without more information

QuestionSelect one

You plot the PDF of a Uniform(0, 0.5) distribution and notice the density is 2 across the whole interval. What does this tell you?

The plot is wrong, because a probability can never exceed 1

The probability of landing in that interval is 2

The density height (2) is fine because what must equal 1 is the area: 2 × 0.5 = 1

The distribution is invalid and should be renormalized

QuestionSelect one

Wait times are Exponential with mean 5 minutes (d = stats.expon(scale=5)). Which expression gives the probability the wait is between 2 and 8 minutes?

d.pdf(8) - d.pdf(2)

d.cdf(8) - d.cdf(2)

d.cdf(8) + d.cdf(2)

d.ppf(8) - d.ppf(2)

QuestionSelect one

You need the value below which 95% of a normal variable's outcomes fall (the 95th percentile). Which scipy method do you use?

d.cdf(0.95)

d.ppf(0.95)

d.sf(0.95)

d.pdf(0.95)

QuestionSelect one

A teammate looks at a density plot, sees the curve peak at a height of 0.6 over the value 50, and concludes "there's a 60% chance the value is 50." What's the precise problem?

They should have read 50, not 0.6, as the probability

Nothing — that's a correct reading of a density plot

The height should have been read off the x-axis instead

A PDF height is a density, not a probability, and the chance of any single exact value is 0; probability is the area over an interval