QBUS5001 Chap.5 Continuous Distributions & the Normal
Continuous Distributions & the Normal
Module 4 covers the continuous distributions used to model business quantities: the Uniform (equal density over an interval), the Exponential (waiting times, rate λ) and, above all, the Normal distribution.
The pivotal skill is Z-standardisation — converting any Normal value to a standard Normal score via Z = (X − μ)/σ — so that one table or one Excel function answers every Normal probability question.
Excel functions NORM.DIST, NORM.S.DIST and their inverses replace the printed tables, but you must still set up the standardisation and read tails correctly.
What this chapter covers
- 01
Uniform distribution: f(x) = 1/(b−a), mean (a+b)/2, variance (b−a)²/12
- 02
Probabilities as areas (proportions of the interval)
- 03
Exponential distribution: rate λ, cdf 1 − e⁻ᴻˣ, mean 1/λ
- 04
Normal distribution N(μ, σ²) and its shape
- 05
Standard Normal Z ~ N(0,1)
- 06
Z-standardisation: Z = (X − μ)/σ
- 07
Excel: NORM.DIST, NORM.S.DIST, NORM.INV, NORM.S.INV
- 08
Reading lower-tail, upper-tail and between probabilities
Uniform distribution mean, variance and a tail probability
- 1 markIdentify a = 8 and b = 20 for the Uniform distribution on [8, 20].
- 1 markExpected value: E[X] = (a+b)/2 = (8+20)/2 = 14 minutes.
- 1 markVariance: Var(X) = (b−a)²/12 = (20−8)²/12 = 144/12 = 12 (minutes²).
- 1 markUpper-tail probability: P(X > 15) = (b − 15)/(b − a) = (20−15)/(20−8) = 5/12.
- 1 markEvaluate: 5/12 = 0.4167. So there is about a 41.67% chance loading exceeds 15 minutes.
Key terms
- Uniform distribution
- A continuous distribution with constant density 1/(b−a) on [a,b]; probabilities are proportions of the interval, mean (a+b)/2, variance (b−a)²/12.
- Exponential distribution
- Models waiting time between events at rate λ: cdf P(X≤x) = 1 − e⁻ᴻˣ, mean 1/λ, variance 1/λ²; memoryless.
- Normal distribution
- The symmetric bell-shaped continuous distribution N(μ, σ²) defined by its mean and variance; the backbone of inference once the CLT applies.
- Standard Normal (Z)
- The Normal distribution with mean 0 and variance 1, N(0,1); any Normal value standardises to a Z-score via Z = (X−μ)/σ.
- Z-standardisation
- Converting a raw value X to a Z-score Z = (X−μ)/σ so that one standard Normal table or function (NORM.S.DIST) gives the probability.
Continuous Distributions & the Normal FAQ
Should I use NORM.DIST or NORM.S.DIST?
NORM.DIST(x, μ, σ, 1) works directly on the raw value and gives the cumulative probability. NORM.S.DIST(z, 1) works on a standardised Z-score. Both give the same answer; standardise first only if the question asks for the Z-score or if you are reading a printed Z-table.
How do I find P(a < X < b) for a Normal?
Compute the two cumulative probabilities and subtract: P(a < X < b) = NORM.DIST(b, μ, σ, 1) − NORM.DIST(a, μ, σ, 1). Sketching the bell and shading the strip prevents subtraction-order errors.
When is the Exponential the right model?
Use it for the time until the next event when events occur at a constant average rate λ — e.g. time between machine breakdowns or customer arrivals. Its rate λ is the reciprocal of the mean waiting time.
Exam move
Master Z-standardisation cold, because it is the gateway to confidence intervals and hypothesis tests in later modules. Always draw the distribution and shade the area you want before computing, and practise translating between raw values and Z-scores in both directions (NORM.INV for the inverse problem of finding a cut-off given a probability).
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