BUSS1020 · Quantitative Business Analysis
Continuous Probability Distributions
Continuous Probability Distributions (Week 5, Berenson Ch 6.1–6.5) shifts from counts to measurements, where probability is the area under a density curve and the chance of any exact value is zero. The star is the normal distribution and its standardised form, the standard normal Z, used with the cumulative table (Table E.2). You also meet the uniform distribution (flat over an interval) and the exponential distribution (time between Poisson events), and learn the four-step procedure for normal probabilities and its inverse.
What this chapter covers
- 01Continuous random variable; probability as area; P(exact) = 0
- 02Uniform distribution: mean (a+b)/2, variance (b−a)²/12
- 03Normal N(μ, σ²): symmetric, bell-shaped, mean = median = mode
- 04Standard normal Z = (X − μ)/σ and Table E.2
- 05Four-step procedure for normal probabilities
- 06Inverse normal: finding X from a percentile
- 07Exponential distribution: mean 1/λ, P(X ≤ x) = 1 − e⁻λx
- 08Poisson ↔ exponential link; Excel (NORM.DIST, NORM.INV, EXPON.DIST)
Normal probability and the inverse normal
- 1 mark(a) Standardise 55: Z = (55 − 45)/8 = 10/8 = 1.25.
- 2 marksFrom Table E.2, P(Z ≤ 1.25) ≈ 0.8944, so P(Z > 1.25) = 1 − 0.8944 = 0.1056.
- 1 mark(a) Conclusion: about 10.6% of deliveries take more than 55 minutes. Excel: =1 − NORM.DIST(55, 45, 8, 1).
- 1 mark(b) The fastest 25% sit below the 25th percentile, so find the Z with cumulative area 0.25: Z ≈ −0.67.
- 1 mark(b) Convert back to minutes: X = μ + Zσ = 45 + (−0.67)(8) = 45 − 5.36 = 39.64 minutes.
- 1 mark(b) Conclusion: the fastest 25% of deliveries arrive in under about 39.6 minutes. Excel: =NORM.INV(0.25, 45, 8).
Key terms
- Probability density
- For a continuous variable, probability is the area under the density curve over an interval; the probability of any single exact value is zero.
- Normal distribution
- A symmetric, bell-shaped distribution N(μ, σ²) where the mean, median and mode coincide and total area under the curve equals 1.
- Standard normal Z
- The normal distribution standardised to mean 0 and standard deviation 1 via Z = (X − μ)/σ, so any normal probability can be read from a single table.
- Uniform distribution
- A continuous distribution that is constant (flat) over an interval [a, b], with mean (a + b)/2 and variance (b − a)²/12.
- Exponential distribution
- Models the waiting time between events occurring at rate λ, with mean 1/λ and CDF P(X ≤ x) = 1 − e⁻λx; it is the continuous partner of the Poisson count.
Continuous Probability Distributions FAQ
Why is the probability of an exact value zero for a continuous variable?
Because probability is area under the curve, and a single point has zero width and hence zero area. This is why for continuous distributions P(X < 55) and P(X ≤ 55) are the same — the boundary point contributes nothing.
How do I use Table E.2 for a 'greater than' or 'between' probability?
Table E.2 gives cumulative probabilities P(Z ≤ z). For P(Z > z) compute 1 minus the table value; for a 'between' probability subtract the smaller cumulative from the larger. Always standardise to Z first.
How are the Poisson and exponential distributions related?
If events occur as a Poisson process at rate λ, then the time BETWEEN consecutive events follows an exponential distribution with the same λ. One counts events in a window; the other measures the gap to the next event.
Exam move
Master the four-step normal procedure until it is reflexive: draw the curve, standardise, read the table, and convert back if needed. Keep clear in your head which direction a question runs — from an X value to a probability (standardise then look up) versus from a probability to an X value (look up the Z then unstandardise). Practise the inverse normal especially, since percentile questions trip students up. Cross-check your by-hand table answers against the Excel NORM.DIST / NORM.INV results while studying, but rehearse the table method for the closed-book exam.
