Probability Distributions
Statistics
Probability Distributions
Syllabus tag: KASNEB CPA | Foundation Level | CA15 Quantitative Analysis | Topic 7 Probability Distributions
Lesson objectives
By the end of this topic, you will be able to:
- Distinguish discrete from continuous distributions
- Apply the binomial distribution and state its conditions
- Apply the Poisson distribution and state when it is appropriate
- Standardise a value and use normal distribution tables
- Explain when one distribution approximates another
Why this matters
Rather than computing probabilities from first principles every time, a distribution supplies a formula for a whole family of situations. Choosing the right distribution is most of the work; the arithmetic follows.
Discrete and continuous
Discrete variables take separate countable values — the number of defective items, customers arriving, faults per machine. Probability attaches to each specific value.
Continuous variables take any value in a range — weight, time, length. The probability of any single exact value is zero, so probability attaches only to intervals.
That is why P(X = 500) is meaningless for a continuous variable while P(490 < X < 510) is not.
The binomial distribution
Conditions, all four of which must hold:
- A fixed number of trials, n
- Each trial has only two outcomes — success or failure
- The probability of success p is constant across trials
- Trials are independent
P(X = r) = ⁿCᵣ × pʳ × (1 − p)ⁿ⁻ʳ
Mean = np Variance = np(1 − p)
Worked example. 10 items are inspected and 20% are defective.
| Working | Result | |
|---|---|---|
| P(exactly 3 defective) | ¹⁰C₃ × 0.2³ × 0.8⁷ | 0.2013 |
| P(none defective) | 0.8¹⁰ | 0.1074 |
| P(at least one) | 1 − 0.1074 | 0.8926 |
| Mean | 10 × 0.2 | 2.0 |
| Standard deviation | √(10 × 0.2 × 0.8) | 1.2649 |
Note the third row. "At least one" is almost always computed as 1 minus "none" — the direct route would require summing ten separate terms.
The independence condition is the one most often broken in practice. Sampling without replacement from a small population changes p at each draw, and the binomial no longer applies exactly.
The Poisson distribution
Used for the number of occurrences of a rare event in a fixed interval of time, space or output — machine breakdowns per week, customers arriving per hour, flaws per metre of cloth.
P(X = r) = e⁻ᵐ × mʳ / r! where m is the mean number of occurrences.
Mean = Variance = m. That equality is a distinctive feature and a useful check: if a dataset's variance is far from its mean, Poisson is the wrong model.
Worked example, with an average of 4 breakdowns a week:
| Working | Result | |
|---|---|---|
| P(no breakdowns) | e⁻⁴ | 0.0183 |
| P(exactly 3) | e⁻⁴ × 4³ / 3! | 0.1954 |
| P(at most 1) | e⁻⁴(1 + 4) | 0.0916 |
Poisson also approximates the binomial where n is large and p is small — conventionally n above 30 and np below 5 — with m = np. It is far quicker than computing large factorials.
:::checkpoint A company records an average of 4 faults a week with a variance of 11. Explain what this suggests about using a Poisson model, and what the data may look like. :::
The normal distribution
Continuous, symmetrical and bell-shaped, defined by its mean and standard deviation. Its properties:
- Mean = median = mode
- About 68% of values lie within 1 standard deviation of the mean
- About 95% within 2 (more precisely 1.96)
- About 99.7% within 3
Standardising. Any normal variable becomes the standard normal by:
z = (x − μ) / σ
A distribution with μ = 500 and σ = 60:
| Value | z | Probability |
|---|---|---|
| 590 | (590 − 500) / 60 = 1.5 | P(Z < 1.5) = 0.9332 |
| P(Z > 1.5) = 0.0668 | ||
| 440 | (440 − 500) / 60 = −1.0 | |
| Between 440 and 590 | 0.9332 − 0.1587 = 0.7745 |
The tables are symmetrical, so P(Z < −1) = P(Z > 1) = 0.1587. Tables usually give only positive z, and using that symmetry is expected.
Sketch the curve and shade the area wanted. Most errors in this topic come from taking the wrong tail rather than from the arithmetic.
Which distribution to use
| Situation | Distribution |
|---|---|
| Fixed number of trials, two outcomes | Binomial |
| Rare events in an interval, no fixed n | Poisson |
| Continuous measurement, symmetrical | Normal |
| Large n, small p | Poisson as an approximation to binomial |
| Large n, p near 0.5 | Normal as an approximation to binomial |
The clue in an examination question is usually the phrasing. "Out of 20 items" signals a fixed n and points to binomial. "On average 4 per hour" signals a rate with no fixed n and points to Poisson.
:::checkpoint A quality inspector examines 200 items from a production line with a 1.5% defect rate. State which distribution applies exactly, which approximation would be acceptable, and why the approximation is worth using. :::