Continuous Distributions

By Victor Arce

Discrete distributions give a probability to each whole-number outcome. But many quantities aren't whole numbers — a waiting time, a height, a measurement error. For those we need continuous distributions, where probability isn't a bar height but the area under a density curve.

Step through the three you'll meet most. The dashed line marks each distribution's mean.

Density, not probability

For a continuous variable, the chance of landing on exactly one value is zero — there are infinitely many values. So instead of "probability of x", we plot a probability density f(x). What carries meaning is the area under f over a range: the probability of landing between a and b is the area from a to b, and the total area under any density is exactly 1.

The three workhorses

• Uniform [a, b]. Every value in the interval is equally likely, so the density is a flat top of height 1/(b − a). It's the "no reason to prefer any value" model — and the source most random-number generators start from.
• Exponential (λ). The waiting time until the next event when events happen at a steady rate λ. It decays from a peak at 0 and is memoryless: having already waited tells you nothing about how much longer you'll wait. Its mean is 1/λ.
• Normal (μ, σ). The bell curve, centered at the mean μ with spread σ. It's everywhere because of the Central Limit Theorem: add up many small, independent effects and their total is approximately normal — the same bell the coin flips filled in.

How they connect to the discrete world

A continuous density is what a discrete histogram becomes in the limit: make the bars narrower and more numerous and the staircase smooths into a curve. The binomial bars from the discrete article settle into the normal curve; the geometric "wait for the first success" becomes the exponential waiting time. Same stories, now on a continuum.

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