Types of Distribution

The agency wants to model three things: whether a flat rents within a week, how many enquiries a listing gets per hour, and how rents vary among similar flats. Each needs a different probability distribution. Choosing the right one is less about formulas and more about recognising the story behind the numbers.

The rule

Name the story first; the distribution follows from it.

Five stories cover most beginner work:

At the agency:

• Bernoulli: one listed flat either rents within a week or not, with $p = 0.6$.
• Binomial: out of 5 listed flats, how many rent within a week? Mean $5 \times 0.6 = 3$. The chance all five do is $0.6^5 \approx 0.078$. For 2 flats, exactly one rents fast with probability $2 \times 0.6 \times 0.4 = 0.48$: two orders, each $0.6 \times 0.4$.
• Poisson: enquiries per hour, averaging $\lambda = 4$. Its variance also equals 4.
• Uniform: last lesson's rents spread evenly from 10 to 30, mean 20.
• Normal: rents of similar two-room flats with mean 20 and standard deviation 4. The 68-95-99.7 rule: about 68% lie within one standard deviation (16 to 24), 95% within two (12 to 28) and 99.7% within three (8 to 32).

The normal curve appears so often because averages and sums of many small independent effects tend towards it. That is also why many ML methods assume normally distributed errors.

The shortcut

Ask two questions: Is it a count or a measurement? If a count, is the number of trials fixed?

The trap is using a binomial when there is no fixed $n$. "How many enquiries today?" has no upper limit, so it is Poisson.

Drills

Answer fast — trust the rule.

Remember

• Name the story first; the distribution follows from it.
• Shortcut: count or measurement? If a count, is n fixed? Fixed means binomial, a rate means Poisson.
• Trap: forgetting the number of orders in a binomial, or forcing a fixed n onto an open-ended count.