Continuous quantities such as time and length are described through interval probabilities.
Waiting time is not restricted to 3 or 4 minutes. It may be 3.2, 3.21, or 3.214 minutes—any value in an interval.
For a continuous variable, probabilities are assigned to intervals rather than exact points.
Key question
Should a continuous process be represented by a uniform, exponential, or normal distribution?
Why are continuous probabilities calculated over intervals?
The height of a probability density curve is not itself a probability. The area under the curve across an interval is.
Probability at one exact point
For a continuous variable, P(X = exactly 3.2) = 0. We calculate probabilities such as P(3 ≤ X ≤ 4).
How do the three distributions differ?
Uniform
Equal density across an interval
random positions · random times
Exponential
Waiting time until the next event
failure · request · arrival gaps
Normal
Symmetric concentration around a mean
measurement error · height · averages
| Distribution | Key parameter | Mean |
|---|---|---|
| Uniform(a,b) | endpoints a and b | (a+b)/2 |
| Exponential(λ) | event rate λ | 1/λ |
| Normal(μ,σ²) | mean μ and standard deviation σ | μ |
Why does the normal distribution appear so often?
Height, measurement error, and repeated-sample averages can look approximately normal because many small effects combine.
| Interval | Approximate normal probability |
|---|---|
| μ ± 1σ | about 68% |
| μ ± 2σ | about 95% |
| μ ± 3σ | about 99.7% |
Normal is not universal
Waiting times, income, and failure intervals often have long right tails and may not be well described by a normal model.
How should a distribution be chosen?
| Situation | Distribution to consider |
|---|---|
| A random point within a fixed interval | Uniform |
| Time until the next request | Exponential |
| A measurement produced by many small effects | Normal |
Memorylessness
For an exponential waiting time, having already waited a long time does not reduce the remaining expected wait.
Change the distribution shapes
Adjust the uniform interval, exponential rate λ, and normal μ and σ.
Key lesson
A continuous distribution represents a data-generating process and probabilities over intervals.