Probability Distributions: Binomial, Poisson & Normal

TAKEAWAYS:

• Binomial distribution models exactly n independent trials with two outcomes (success/failure) and constant probability p (e.g., pass/fail, yes/no).
• Poisson distribution describes rare events in a fixed interval (e.g., customer arrivals, defects per unit) with λ = mean = variance.
• Normal distribution is bell-shaped, symmetric, and used for continuous data (e.g., heights, exam scores, stock returns) with 68-95-99.7 rule.
• Key formulas:
  • Binomial: $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$, $E(X)=np$, $\text{Var}(X)=np(1-p)$.
  • Poisson: $P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$, $E(X)=\text{Var}(X)=\lambda$.
  • Normal: $P(a\le X\le b)=\Phi \left(\frac{b-\mu }{\sigma }\right)-\Phi \left(\frac{a-\mu }{\sigma }\right)$.
• Applications: Binomial for quality control (defective items), Poisson for call center arrivals, Normal for financial risk modeling.
• Exam focus: Calculate probabilities, find parameters (n, p, λ), and interpret real-world scenarios (e.g., loan defaults, delivery delays).

1. Binomial Distribution: Counting Successes in Fixed Trials

Definition and Key Features

The binomial distribution models the number of successes (X) in n independent trials, where each trial has:

• Two possible outcomes: success (probability p) or failure (probability 1-p).
• Constant p across trials (e.g., coin flips, pass/fail exams).
• Trials are independent (outcome of one does not affect others).

Notation:

• $X\sim B(n,p)$: Binomial with n trials and success probability p.
• PMF: $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$, where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$.

Worked Example 1: Probability of Defective Items (Binomial)

Scenario: A factory produces light bulbs with a 5% defect rate (p=0.05). If a quality inspector checks 20 bulbs, what is the probability that exactly 2 are defective?

Solution: Here, $X\sim B(20,0.05)$, and we want $P(X=2)$. $$P(X=2)=\left(\genfrac{}{}{0ex}{}{20}{2}\right)(0.05{)}^2(0.95{)}^{18}$$ $$=190\times 0.0025\times 0.3774\approx 0.1807\text{ or }18.07\%.$$

Real-World Tie-In:

• Daraz Quality Control: Daraz uses binomial distribution to estimate the probability of defective orders in a batch of 1000 items with a 2% defect rate. For example, $P(X=5)$ helps decide whether to reject a shipment.

2. Poisson Distribution: Modeling Rare Events

Definition and Key Features

The Poisson distribution models the number of rare events occurring in a fixed interval (time/space) with:

• Constant mean rate λ (e.g., arrivals per hour, defects per km).
• Events occur independently.
• Mean = Variance = λ.

Notation:

• $X\sim \text{Poisson}(\lambda )$.
• PMF: $P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$.

Worked Example 2: Call Center Arrivals (Poisson)

Scenario: A call center receives an average of 3 calls per minute (λ=3). What is the probability of exactly 5 calls in the next minute?

Solution: $$P(X=5)=\frac{e^{-3}{3}^5}{5!}=\frac{2.207\times 243}{120}\approx 0.1008\text{ or }10.08\%.$$

Real-World Tie-In:

• Pathao Ride Requests: Pathao uses Poisson distribution to predict the number of ride requests per hour in a zone. If λ=10 requests/hour, $P(X=15)$ helps allocate drivers dynamically.
• NTC Power Outages: NTC tracks outages per district using Poisson to plan maintenance (e.g., λ=0.5 outages/day in Kathmandu).

3. Normal Distribution: The Bell Curve

Definition and Key Features

The normal distribution is a continuous distribution:

• Bell-shaped, symmetric about the mean (μ).
• 68-95-99.7 Rule:
  • 68% of data within μ ± σ.
  • 95% within μ ± 2σ.
  • 99.7% within μ ± 3σ.
• PDF: $f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$.

Standard Normal (Z-Score): Convert any normal to $Z\sim N(0,1)$ using: $$Z=\frac{X-\mu }{\sigma }.$$

Worked Example 3: Loan Approvals (Normal)

Scenario: Bank loans have mean μ=500,000 Rs and standard deviation σ=50,000 Rs. What percentage of loans are between 450,000 Rs and 550,000 Rs?

Solution: Convert to Z-scores: $$Z_1=\frac{450,000-500,000}{50,000}=-1,Z_2=\frac{550,000-500,000}{50,000}=1.$$ From Z-table: $$P(-1\le Z\le 1)=\Phi (1)-\Phi (-1)=0.8413-0.1587=0.6826\text{ or }68.26\%.$$

Real-World Tie-In:

• Nepal Rastra Bank (NRB) Interest Rates: NRB assumes loan defaults follow a normal distribution to set risk reserves. For example, if μ=3% default rate and σ=0.5%, $P(X>4\%)$ estimates high-risk scenarios.
• NEPSE Stock Returns: Stock analysts use normal distribution to model daily returns (e.g., μ=0.2%, σ=1.5%) and calculate probabilities of gains/losses.

4. Comparing the Three Distributions

5. When to Use Which Distribution?

flowchart TD
    A["Start"] --> B{"Is data discrete or continuous?"}
    B -->|"Discrete"| C{"Are there fixed trials with two outcomes?"}
    C -->|"Yes"| D["Use Binomial"]
    C -->|"No"| E{"Are events rare and independent?"}
    E -->|"Yes"| F["Use Poisson"]
    E -->|"No"| G["Use other discrete distributions"]
    B -->|"Continuous"| H{"Is data symmetric and bell-shaped?"}
    H -->|"Yes"| I["Use Normal"]
    H -->|"No"| J["Use other continuous distributions"]

Key Rules of Thumb:

1. Binomial: Use when you have yes/no outcomes in fixed trials (e.g., pass/fail, defective/non-defective).
2. Poisson: Use for counts of rare events (e.g., accidents, customer complaints).
3. Normal: Use for continuous data (e.g., heights, test scores, stock prices).

6. Relationships Between Distributions

• Poisson Approximates Binomial: If $n$ is large and $p$ is small (e.g., $n=100$, $p=0.01$), then $B(n,p)\approx \text{Poisson}(np)$. Example: $B(100,0.01)\approx \text{Poisson}(1)$.

• Normal Approximates Binomial: If $n$ is large (e.g., $n\ge 30$), then $B(n,p)\approx N(np,\sqrt{np(1-p)})$. Example: $B(50,0.5)\approx N(25,\sqrt{12.5})$.

• Poisson Approximates Normal: If $\lambda$ is large (e.g., $\lambda \ge 10$), then $\text{Poisson}(\lambda )\approx N(\lambda ,\sqrt{\lambda })$. Example: $\text{Poisson}(15)\approx N(15,\sqrt{15})$.

7. Solving Past Exam Questions

Question 1: Binomial Probability (TU 2078)

A bag contains 20 balls numbered 1 to 20. A ball is drawn at random without replacement. What is the probability that the number is a multiple of 3 or 7?

Solution:

• Total balls: 20.
• Multiples of 3: 3, 6, 9, 12, 15, 18 → 6 balls.
• Multiples of 7: 7, 14 → 2 balls.
• Multiples of both (21): None (since max is 20).
• Probability: $P(3\text{ or }7)=P(3)+P(7)=\frac{6}{20}+\frac{2}{20}=\frac{8}{20}=0.4$.

Note: This is not binomial because it’s without replacement (hypergeometric distribution). For binomial, trials must be independent and with replacement.

Question 2: Poisson PMF (TU 2077)

Find $P(X=4)$ for a Poisson distribution with $\lambda =1.5$.

Solution: $$P(X=4)=\frac{e^{-1.5}(1.5{)}^4}{4!}=\frac{0.2231\times 5.0625}{24}\approx \mathrm{0.0458.}$$

In the Real World

1. Khalti Transaction Failures:

   • Khalti models failed transactions as a Poisson process with $\lambda =0.05$ failures per 100 transactions. For 500 transactions, $P(X=3)$ helps estimate refunds.

2. Ncell Customer Complaints:

   • Ncell tracks complaints per day using Poisson with $\lambda =10$. $P(X>15)$ triggers extra customer support staff.

3. Daraz Delivery Delays:

   • Daraz assumes delivery delays follow a normal distribution with $\mu =2$ days and $\sigma =0.5$ days. $P(X>3)$ estimates late penalties.

Exam Tip

1. Identify the Distribution:

   • Discrete + fixed trials + two outcomes → Binomial.
   • Discrete + rare events → Poisson.
   • Continuous + symmetric → Normal.

2. Memorize Key Formulas:

   • Binomial PMF: $\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$.
   • Poisson PMF: $\frac{e^{-\lambda }{\lambda }^k}{k!}$.
   • Normal Z-score: $Z=\frac{X-\mu }{\sigma }$.

3. Practice Z-Table Lookups:

   • Always convert to standard normal ($Z$) for normal distribution problems.

4. Watch for Tricks:

   • Without replacement → Not binomial (use hypergeometric).
   • Large n, small p → Poisson approximation.
   • Large n → Normal approximation.

5. Real-World Scenarios:

   • Binomial: Quality control (defective items), pass rates.
   • Poisson: Customer arrivals, machine failures.
   • Normal: Heights, exam scores, financial returns.

Standard normal distribution (μ=0, σ=1) with shaded 68% area (Image: ssindhwani, CC BY-SA 4.0, via Wikimedia Commons)