Discrete Probability Distributions – Binomial & Poisson

1. Introduction to Discrete Distributions

A discrete random variable $X$ takes values from a countable set $\{x_1,x_2,\dots \}$. Its probability distribution is fully described by the probability mass function (pmf)

$$p_X(x)=P(X=x),\sum _x{p}_X(x)=1.$$

Two of the most frequently used discrete distributions in engineering, finance, and everyday life are the binomial and Poisson distributions.

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2. Binomial Distribution

2.1 Definition & Conditions

A random variable $X$ follows a binomial distribution, written $X\sim \text{Bin}(n,p)$, when the following four conditions hold:

2.2 Probability Mass Function

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

• $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$ counts the number of ways to choose $k$ successes out of $n$ trials.

2.3 Mean, Variance & Standard Deviation

$${\mu }_X=E[X]=np,{\sigma }_X^2=Var(X)=np(1-p),{\sigma }_X=\sqrt{np(1-p)}.$$

2.4 Worked Example – eSewa Transaction Success

Problem: eSewa reports that 2 % of its online transactions fail due to network glitches. If a merchant processes 20 transactions in a day, what is the probability that exactly 2 transactions fail?

Solution

• Here a “failure’’ is a “success’’ for the binomial model (we count failures).
• $n=20,p=0.02,k=2$.

$$\begin{array}{rl}P(X=2)& =\left(\genfrac{}{}{0ex}{}{20}{2}\right)(0.02{)}^2(0.98{)}^{18}\\ & =190\times 0.0004\times 0.6983\\ & \approx \mathrm{0.0531.}\end{array}$$

So there is a 5.3 % chance that exactly two transactions fail on that day.

2.5 Normal Approximation (Large n)

When $n$ is large and $np(1-p)\ge 5$, the binomial can be approximated by a normal distribution

$$X\approx N(np,np(1-p))$$

with a continuity correction of $\pm 0.5$.

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3. Poisson Distribution

3.1 Definition & Conditions

A random variable $Y$ follows a Poisson distribution, written $Y\sim \text{Pois}(\lambda )$, when it counts the number of rare, independent events occurring in a fixed interval (time, area, volume, etc.). The conditions are:

3.2 Probability Mass Function

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

3.3 Mean & Variance

$${\mu }_Y=E[Y]=\lambda ,{\sigma }_Y^2=Var(Y)=\lambda .$$

Both mean and variance are equal to the rate parameter $\lambda$.

3.4 Worked Example – Ncell Call Arrivals

Problem: Ncell observes an average of 12 call attempts per minute on its customer‑service line. Assuming a Poisson model, what is the probability that exactly 8 calls arrive in a particular minute?

Solution

• $\lambda =12,k=8$.

$$\begin{array}{rl}P(Y=8)& =\frac{e^{-12}12^8}{8!}\\ & =\frac{e^{-12}429981696}{40320}\\ & \approx \mathrm{0.094.}\end{array}$$

Thus, there is a 9.4 % chance of receiving exactly eight calls in a minute.

3.5 Poisson Approximation to Binomial

When $n$ is large, $p$ is small, and $np=\lambda$ is moderate, the binomial $\text{Bin}(n,p)$ can be approximated by $\text{Pois}(\lambda )$.

Example: In a Daraz warehouse, the daily defect rate of a certain electronic component is 0.3 % (p = 0.003). For a batch of 500 items (n = 500), $\lambda =np=1.5$. The probability of finding exactly 2 defective items can be computed using the Poisson pmf with $\lambda =1.5$.

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4. Comparison of Binomial & Poisson

Advantages / Disadvantages

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5. Applications in Nepal

Real‑World Worked Example – Daraz Order Queue

Daraz receives on average 30 orders per hour for a popular gadget. The probability that exactly 5 orders arrive in a 10‑minute window is needed to schedule pick‑pack staff.

• Convert rate: ${\lambda }_{10\text{min}}=30\times \frac{10}{60}=5$.
• Use Poisson with $\lambda =5$:

$$P(Y=5)=\frac{e^{-5}{5}^5}{5!}=\mathrm{0.175.}$$

Thus, there is a 17.5 % chance that exactly five orders will appear in any ten‑minute slot, guiding staffing decisions.

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6. Step‑by‑Step Procedure (Mermaid)

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7. Common Mistakes to Avoid

1. Confusing “success’’ with “failure’’ – always define which outcome you are counting.
2. Using binomial when $p$ changes – e.g., defect rate that varies across production lines requires a mixture model, not a simple binomial.
3. Forgetting the continuity correction when applying the normal approximation to a binomial.
4. Applying Poisson to high‑rate events – when λ > 30, a normal approximation is more accurate.

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8. In the Real World

• eSewa: When a merchant processes 50 payments, the probability that more than 2 fail is computed with a binomial model (n = 50, p = 0.02). This informs risk‑mitigation fees.
• Ncell: Call‑arrival data collected over a month shows an average of 9 calls per minute. Ncell uses a Poisson model to predict staffing needs; a sudden spike to 20 calls in a minute triggers an automatic queue‑expansion algorithm.
• Daraz: During a flash sale, order arrivals surge to an average of 120 orders per hour. The Poisson model (λ = 2 per minute) helps the logistics team allocate additional delivery vans in real time.

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9. Summary

• Binomial and Poisson are the cornerstone discrete distributions for modelling counts.
• Correct identification of the underlying process (fixed trials vs. rare events) determines which distribution to use.
• Both have simple closed‑form pmfs, easy‑to‑remember mean/variance formulas, and useful normal approximations for large parameters.
• Real‑world Nepalese services (eSewa, Ncell, Daraz) rely on these models for operational decisions, risk assessment, and resource planning.

Exam tip

• Definition first: Write the pmf, list the four binomial conditions, and the Poisson assumptions.
• Formula sheet: Memorise $E[X]=np$, $Var(X)=np(1-p)$ for binomial; $E[Y]=\lambda$, $Var(Y)=\lambda$ for Poisson.
• Worked‑example pattern: Identify n & p (or λ), plug into the pmf, simplify, and if required, use a calculator for factorials or exponentials.
• Approximation: If the question mentions “large n” or “rare events”, state the appropriate normal approximation and apply the continuity correction.
• Comparison table: A quick 2‑row table (as shown above) earns marks for clarity.

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