Business Statistics – Discrete Distributions (MDB2013 Chapter 4)

Teaching & Learning Outcomes

• Compute probabilities from various discrete distributions.
• Utilize discrete probability plots to determine financial applications.
• Calculate probabilities specifically for the Binomial, Poisson, and Hypergeometric distributions.

Probability Distribution – Definition & Characteristics

• Definition: A listing of all experiment outcomes with their associated probabilities.
• Key characteristics:
  • Probabilities lie in the interval $[0,1]$.
  • Outcomes are mutually exclusive.
  • Outcomes are exhaustive → $\sum P(x)=1$.

Illustrative Example: 3 Coin Tosses

• Experiment: Count heads when a coin is tossed 3 times.
• Possible values of $X$: $0,1,2,3$ heads.
• Probabilities:
  • $P(0)=\frac{1}{8}=0.125$
  • $P(1)=\frac{3}{8}=0.375$
  • $P(2)=\frac{3}{8}=0.375$
  • $P(3)=\frac{1}{8}=0.125$
• Check: $0.125+0.375+0.375+0.125=1.0$ (distribution validated).

Random Variables

• Outcome values produced by a chance experiment.
• Examples:
  • Number of employees absent on Monday (quantitative discrete).
  • Grade level of team members (qualitative categorical variable but treated as a random variable).

Types

• Discrete: Assumes clearly separated values (e.g., number of credit cards a customer holds).
• Continuous: Infinite possible values in an interval (e.g., flight time KLIA–Langkawi; snowfall depth).

Mean, Variance & Standard Deviation of a Discrete Distribution

• Mean (Expected value): $\mu = \sum xP(x)$.
• Variance: $\sigma^{2}= \sum (x-\mu)^{2}P(x)$.
• Standard deviation: $\sigma = \sqrt{\sigma^{2}}$.

Example: Johan’s Car Sales on Saturday

• Distribution (condensed): $P(0)=0.1,\ P(1)=0.2,\ P(2)=0.3,\ P(3)=0.3,\ P(4)=0.1$.
• Calculations:
  • $\mu = 2.1$ cars.
  • Tabled variance computation (\sum (x-\mu)^2 P(x)) → $\sigma^{2}=1.29$.
  • Interpretation: On a typical Saturday Johan expects about $2.1$ cars sold with moderate variability (SD $\approx1.136$).

Binomial Distribution

• Definition: Widely occurring discrete distribution meeting 4 criteria:
  1. Only two outcomes (success/failure).
  2. Fixed, known number of trials $n$.
  3. Constant success probability $\pi$ across trials.
  4. Trials are independent.
• Probability mass function (PMF):
  $P(X=x)=C^{n}_{x}\pi^{x}(1-\pi)^{n-x}$ for $x=0,1,\dots,n$.
• Mean & Variance:
  • $\mu = n\pi$
  • $\sigma^{2}=n\pi(1-\pi)$

Debit-Card Coffee-Shop Example (Zus Coffee, $n=5,\pi=0.28$)

• (a) Exactly one debit-card purchase:
  $P(1)=C^{5}_{1}(0.28)^{1}(0.72)^{4}$.
• (b) Full distribution: Evaluate PMF for $x=0\rightarrow5$.
• (c) $P(X\ge 3)=P(3)+P(4)+P(5)$.
• (d) $P(X\le 2)=1-P(X\ge3)$ (or directly sum).
• (e) $\mu =5(0.28)=1.4,$ $\sigma^{2}=5(0.28)(0.72)=1.008$.
• Verify all 4 binomial requirements (yes – meets each).

Binomial Tables Example (Dropped Calls, $n=6,\pi=0.05$)

• Use Table 6-2.
  • (a) $P(0)=0.7351$ (value from table).
  • (b) $P(1)=0.2326$, etc. through $P(6)=1.6\times10^{-8}$.

Hypergeometric Distribution

• Used when sampling without replacement from a finite population where n/N>0.05 (lack of independence).
• Conditions:
  1. Two outcome categories.
  2. Fixed sample size $n$.
  3. Trials are dependent.
  4. Population finite and sampling w/out replacement.
• PMF: $P(X=x)=\dfrac{C^{S}<em>{x}\,C^{N-S}</em>{n-x}}{C^{N}_{n}}$.
  • $N$ = population size, $S$ = total successes in population.

Classroom Committee Example

• $N=50,\ S=40,\ n=5,\ x=4$.
• Probability:
  $P(4)=\dfrac{C^{40}<em>{4}\,C^{10}</em>{1}}{C^{50}_{5}}$ (compute via combinations).

Poisson Distribution

• Describes number of occurrences in a specified interval (time, distance, area, volume).
• Assumptions:
  1. Probability proportional to interval length.
  2. Intervals are independent.
• PMF (parameter $\lambda$ = mean number of events per interval):
  $P(X=x)=\dfrac{\lambda^{x}e^{-\lambda}}{x!}$.
• Relation to Binomial: Limiting case when $n\to\infty$, $\pi\to0$, $n\pi=\lambda$.

Lost-Bags Example (Amal by Malaysia Airlines)

• Sample: 500 flights, 20 lost bags → overall rate $\lambda_{500}=20$.
• (a) Valid Poisson: counting occurrence per non-overlapping flight, independence presumed.
• (b) Mean per flight: $\lambda =20/500=0.04$ bags.
• (c) No bags lost on a flight: $P(0)=e^{-0.04}=0.9608$.
• (d) At least one lost: $1-P(0)=0.0392$.

Poisson Table Example (Truck Breakdowns)

• Mean breakdowns per run: $\lambda=0.30$.
• From table:
  • $P(0)=0.7408$
  • $P(1)=0.2222$ (matches $\lambda e^{-\lambda}$ check).

Conceptual Connections & Applications

• Financial modeling often uses binomial trees for option pricing; discrete plots help visualize risk.
• Quality-control auditing (hypergeometric) models defect sampling without replacement.
• Operations management (Poisson) forecasts arrival rates—e.g., call centers, traffic, insurance claims.
• Ethical note: Accurate statistical modeling prevents misallocation of resources, fosters fair decisions.

Formulas Consolidated

• Discrete Mean: $\mu=\sum xP(x)$
• Discrete Variance: $\sigma^{2}=\sum (x-\mu)^{2}P(x)$
• Binomial PMF: $P(X=x)=C^{n}_{x}\pi^{x}(1-\pi)^{n-x}$
• Binomial Mean/Var: $\mu=n\pi,\ \sigma^{2}=n\pi(1-\pi)$
• Hypergeometric PMF: $P(X=x)=\dfrac{C^{S}<em>{x}C^{N-S}</em>{n-x}}{C^{N}_{n}}$
• Poisson PMF: $P(X=x)=\dfrac{\lambda^{x}e^{-\lambda}}{x!}$

Study Tips

• Always check distribution requirements before applying a formula.
• For small $n$ without replacement, prefer hypergeometric; for large $n$ with small $\pi$, Poisson may approximate Binomial.
• Use tables or software to avoid factorial/combinatorial arithmetic mistakes.
• Draw probability plots to see skewness and assess variance vs. mean relationships.