Probability Distributions – Discrete & Continuous, Binomial, Poisson, Normal, Applications

1. What is a Probability Distribution?

A probability distribution describes how the probabilities are spread over the possible values of a random variable $X$.

• Discrete random variable – takes countable values (e.g., number of customers arriving in an hour).
• Continuous random variable – can take any value in an interval (e.g., daily sales amount).

The two must satisfy:

$$\sum _{x}P(X=x)=1\text{(discrete)}\text{or}{\int }_{-\infty }^{\infty }f(x)dx=1\text{(continuous)}$$

Visual: Discrete Probability Mass Function

Visual: Continuous Probability Density Function

2. Discrete Probability Distributions

2.1 Binomial Distribution

Definition – The number of successes $X$ in $n$ independent Bernoulli trials, each with success probability $p$.

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k},k=0,1,\dots ,n$$

• Mean: $\mu =np$
• Variance: ${\sigma }^2=np(1-p)$

Worked Example – Quality Control at a Daraz Warehouse

Daraz inspects 20 randomly selected parcels each day. Historical data show a 5 % defect rate ($p=0.05$).

Question: Probability that exactly 2 parcels are defective.

$$P(X=2)=\left(\genfrac{}{}{0ex}{}{20}{2}\right)(0.05{)}^2(0.95{)}^{18}$$

$$\left(\genfrac{}{}{0ex}{}{20}{2}\right)=190,(0.05{)}^2=0.0025,(0.95{)}^{18}\approx 0.377$$

$$P(X=2)=190\times 0.0025\times 0.377\approx 0.179(17.9\%)$$

Interpretation: On a typical day, Daraz can expect about 18 % chance of finding exactly two defective parcels.

2.2 Poisson Distribution

Definition – Models the number of events occurring in a fixed interval when events are rare and occur independently.

$$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!},k=0,1,2,\dots$$

• $\lambda$ = average number of events per interval (mean = variance = $\lambda$)

Worked Example – Call Center Arrivals (NTC)

NTC’s technical support receives on average 12 calls per hour ($\lambda =12$).

Question: Probability of receiving exactly 8 calls in the next hour.

$$P(X=8)=\frac{e^{-12}12^8}{8!}$$

$$e^{-12}\approx 6.14\times 10^{-6},12^8=429981696,8!=40320$$

$$P(X=8)=\frac{6.14\times 10^{-6}\times 429981696}{40320}\approx 0.094(9.4\%)$$

Interpretation: Roughly a 9 % chance that the call volume will be exactly 8 in an hour, useful for staffing decisions.

Comparison Table

3. Continuous Probability Distributions

3.1 Normal Distribution

Definition – A continuous distribution with density

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left[-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right],-\infty <x<\infty$$

• Mean ($\mu$) – centre of the curve.
• Standard deviation ($\sigma$) – spread; about 68 % of observations lie within $\mu \pm \sigma$.

Worked Example – Bank Loan Interest (Nabil Bank)

A bank reports that the annual interest rate on personal loans follows $N(\mu =12\%,\sigma =2\%)$.

Question: What proportion of loans have interest rates between 10 % and 14 %?

Convert to Z‑scores:

$$z_1=\frac{10-12}{2}=-1,z_2=\frac{14-12}{2}=1$$

Using the standard normal table, $P(-1<Z<1)=0.6826$.

Interpretation: Approximately 68 % of personal loans fall in the 10 %–14 % interest range.

3.2 Other Continuous Distributions (Brief)

4. Selecting the Right Distribution

flowchart LR
    A["Identify variable type"] --> B["Discrete?"]
    B -->|"Yes"| C["Count of successes → Binomial?"]
    C -->|"Fixed n, p"| D["Use Binomial"]
    C -->|"Rare events, large n"| E["Use Poisson"]
    B -->|"No"| F["Continuous?"]
    F -->|"Symmetric bell‑shape"| G["Use Normal (or approximate)"]
    F -->|"Time between events"| H["Use Exponential"]

Guidelines

1. Nature of data – Count vs measurement.
2. Sample size – Small $n$ may need exact binomial; large $n$ with small $p$ → Poisson.
3. Shape – Skewed vs symmetric; normal is appropriate when the Central Limit Theorem applies.

5. Applications in Business

• Inventory Management (Daraz) – Poisson model predicts daily order arrivals, helping set reorder points.
• Call Centre Staffing (NTC, Ncell) – Binomial for success/failure of call resolutions; Poisson for call volume forecasting.
• Risk Assessment (Banks) – Normal distribution for loan interest rates, credit scores, and market returns.
• Quality Control (Manufacturing) – Binomial for defect counts in a batch; control charts rely on normal approximation.

6. Worked Example Integrated with Real Situation

Scenario: A Nepali e‑commerce platform (Khalti) processes online payments. Historical data show an average of 30 successful transactions per hour. Management wants to know the probability that in a particular hour they will process more than 40 transactions.

Because transactions are independent and occur at a relatively constant average rate, we model them with a Poisson distribution ($\lambda =30$).

$$P(X>40)=1-P(X\le 40)=1-\sum _{k=0}^{40}\frac{e^{-30}30^k}{k!}$$

Using a calculator or software:

$$\sum _{k=0}^{40}\frac{e^{-30}30^k}{k!}\approx 0.952$$

$$P(X>40)=1-0.952=0.048(4.8\%)$$

Business implication: Only about 5 % of hours will exceed 40 transactions, so the server capacity can be sized accordingly, with occasional scaling during peak periods.

7. Real‑World Connections

In the real world

1. Daraz order queue – The number of orders arriving per minute follows a Poisson distribution; the probability of 5 orders in a minute helps decide how many pick‑pack staff to schedule.
2. NTC call arrivals – Daily call volume is modeled with a binomial distribution where each incoming call is a “success” (resolved) with probability 0.85; this guides training needs.
3. Bank loan interest rates – As shown in the example, the spread of interest rates is approximated by a normal distribution, allowing banks to price risk‑adjusted products.

IMAGE: "Daraz warehouse workers" | Workers sorting parcels at Daraz’s fulfillment centre

IMAGE: "NTC call centre" | NTC technical support agents handling customer calls

IMAGE: "bank loan documents" | Nabil Bank loan agreement showing interest rate terms

8. Summary of Formulas

Exam tip

• Memorise the key formulas for binomial, Poisson, and normal PDFs/PMFs; the exam often asks to write them directly.
• Identify the situation: look for keywords – “fixed number of trials” → binomial; “rare events in a time/space interval” → Poisson; “symmetrical bell‑shaped” or “average of many independent factors” → normal.
• Use the Z‑score table efficiently: convert any normal variable to the standard normal, then read $P(Z<z)$ or $P(a<Z<b)$.
• When n is large and p is small, remember the Poisson approximation to the binomial (λ = np). Write the condition $n\ge 20,p\le 0.05$.
• Show all steps: state the distribution, plug numbers, compute intermediate values (e.g., $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, $e^{-\lambda }$), and give the final probability to 4 decimal places. Partial credit is awarded for correct setup even if arithmetic slips.