MGMT90280 · Managerial Decision Analytics
Probability Distributions
Probability distributions is the Week-4 topic of MGMT90280 Managerial Decision Analytics at the University of Melbourne, and the foundation for every “decision under uncertainty” model in the subject. It covers discrete vs continuous random variables, the binomial and Poisson (discrete) and uniform, normal and exponential (continuous) distributions, z-scores, and the Excel functions BINOM.DIST, POISSON.DIST, NORM.DIST, NORM.INV and EXPON.DIST. These same distributions are what the Monte-Carlo simulation question later samples from.
What this chapter covers
- 01Random variables: discrete (mass function f(x), countable values) vs continuous (density f(x), where P(X = a single value) = 0)
- 02Binomial(n, p): number of successes in n independent trials; mean np, variance np(1−p); Excel BINOM.DIST
- 03Poisson(μ): count of events per interval; mean = variance = μ; Excel POISSON.DIST
- 04Uniform(a, b): equally-likely values; E = (a+b)/2, Var = (b−a)²/12
- 05Normal(μ, σ): symmetric bell; z-score z = (x−μ)/σ standardises any value
- 06Exponential (mean μ): time between random events; cdf 1−e^(−x/μ); Excel rate = 1/μ
- 07Excel toolkit: NORM.DIST (left tail), NORM.INV (percentile / inverse), the cumulative TRUE/FALSE flag
- 08Exact vs cumulative: FALSE gives P(X = x), TRUE gives P(X ≤ x); upper tail = 1 − cdf
- 09Discrete ≤ vs < care, and the exponential rate = 1/mean trap
Normal handling time, a z-score, and a service-level decision
- +1(a) Standardise, then read the left tail. z = (6 − 4.0)/1.2 = 1.667, so P(X < 6) = P(Z < 1.667) = NORM.DIST(6, 4, 1.2, TRUE) = 0.9522.
- +1(b) Upper tail via the complement. z = (5 − 4.0)/1.2 = 0.833, so P(X > 5) = 1 − NORM.DIST(5, 4, 1.2, TRUE) = 1 − 0.7977 = 0.2023.
- +1(c) 90th percentile = inverse normal. The z with P(Z ≤ z) = 0.90 is 1.2816, so x = 4.0 + 1.2816·1.2 = NORM.INV(0.90, 4, 1.2) = 5.54 min.
- +1(d) Decision. The target needs P(X ≤ 6) ≥ 0.95. From (a), P(X ≤ 6) = 0.9522 ≥ 0.95, so the 95%-within-6-minutes target IS met.
Key terms
- Random variable
- A numerical outcome of a random experiment. Discrete if it takes countable values (described by a probability mass function); continuous if it takes values over an interval (described by a probability density).
- Probability mass function (pmf)
- For a discrete variable, f(x) = P(X = x); the masses are non-negative and sum to 1. Excel returns it with the cumulative flag FALSE.
- Probability density function (pdf)
- For a continuous variable, a curve f(x) whose area over an interval gives the probability of that interval; total area = 1 and P(X = any single value) = 0.
- Binomial distribution
- Counts successes in n independent identical trials each with success probability p: f(x) = C(n,x) p^x (1−p)^(n−x), mean np, variance np(1−p). Excel BINOM.DIST(x, n, p, cumulative).
- Poisson distribution
- Counts events in a fixed interval at constant rate μ: f(x) = μ^x e^(−μ)/x!, with mean = variance = μ. Excel POISSON.DIST(x, μ, cumulative).
- z-score
- The standardised value z = (x − μ)/σ, i.e. how many standard deviations x lies from the mean. It lets one function serve every normal distribution.
- NORM.INV
- The inverse-normal Excel function: NORM.INV(p, μ, σ) returns the value x for which P(X ≤ x) = p, i.e. a percentile. The reverse of NORM.DIST.
- Exponential distribution
- Models the time between independent random events (mean μ): P(X ≤ x) = 1 − e^(−x/μ). In Excel EXPON.DIST the second argument is the rate λ = 1/μ, not the mean.
Probability Distributions FAQ
When do I use the binomial versus the Poisson?
Use the binomial when you have a fixed number of trials n, each a success/failure with the same probability p, and you count successes (e.g. how many of 15 applications are declined). Use the Poisson when you count events over an interval at an average rate μ with no fixed number of trials (e.g. calls per hour); its giveaway is that the mean equals the variance.
What is the difference between NORM.DIST and NORM.INV?
NORM.DIST(x, mean, SD, TRUE) takes a value x and returns the cumulative probability to its left, P(X ≤ x). NORM.INV(p, mean, SD) does the reverse: give it a probability p and it returns the value x (a percentile). For an upper tail, take 1 − NORM.DIST; for a between-values probability, subtract two NORM.DIST calls.
Can AI help me with probability distributions in MGMT90280?
Yes, for understanding. Sia can explain step by step which distribution fits a scenario, define every symbol, and walk you through the matching Excel function and the cumulative TRUE/FALSE flag, so you can reproduce the working yourself. It will not sit an assessment for you or promise a grade — and note that Assignment 1 runs in the Respondus LockDown Browser with no internet or generative AI, so use Sia only to learn beforehand and confirm the rules on the LMS.
Exam move
Master the discrete-vs-continuous split first, because it decides which formula and which Excel function you are allowed to use, and whether ≤ differs from < (it does for discrete, not for continuous). Build a one-line card for each of the five distributions: its use-case, its mean and variance, and its Excel call with the exact argument order. Drill the four recurring traps — the cumulative TRUE/FALSE flag (FALSE = exact, TRUE = ≤), taking 1 − cdf for an upper tail, EXPON.DIST wanting the rate 1/mean, and Uniform variance being (b−a)²/12. This topic is assessed directly in Assignment 1 (a timed Respondus quiz) and underpins the Monte-Carlo question in the final exam, so practise both by-hand z-scores and the Excel functions. Budget time in proportion to marks (about 1.2 minutes per mark), and always show the formula or Excel call before the number. Confirm all assessment dates and rules on the LMS.
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