Track 5 · Probability Distributions · Lesson 5 of 10

Uniform and exponential distributions

Calculate probabilities, means and standard deviations for the continuous uniform and exponential distributions, and judge when an exponential waiting-time model is reasonable.

30 minIntermediate8 graded exercisesMembers

A contact centre needs to know how often two calls arrive within seconds of each other, and a commuter wants to know how long they might stand at a bus stop. Both are questions about waiting times — continuous quantities whose probabilities are areas under a curve.

A continuous random variable can take any value in an interval: 3.2 minutes, 3.21 minutes, 3.2147 minutes… Its distribution is described by a probability density function (pdf) $f(x)$ . Probabilities are areas under the density:

P(a \le X \le b) = \text{area under } f \text{ between } a \text{ and } b.

A single point has no width, so for any continuous variable $P(X = x) = 0$ . That is why, for continuous variables, $P(X \le x)$ and $P(X < x)$ are the same number — unlike the counts of the earlier lessons.

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Compute uniform probabilities as rectangle areas, with mean (a+b)/2 and SD (b−a)/√12
Use the exponential rate λ (not the mean) in P(T > t) = e^(−λt), and convert between rate and mean
Explain memorylessness and recognise when it is unrealistic
Link Poisson counts to exponential gaps and estimate a rate from data

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