Probability Distributions & Stochastic Processes: Types, Models & Applications

TAKEAWAYS:

• Distributions are mathematical functions describing random variables: discrete (binomial, Poisson) vs. continuous (exponential, normal) with distinct PDF/CDF shapes and use cases.
• Stochastic processes model systems evolving over time (e.g., Markov chains for state transitions, renewal processes for repeated events).
• Key parameters (mean, variance, λ, μ) define distribution behavior and are critical for simulation inputs.
• Real-world ties: Poisson models customer arrivals at Daraz, exponential models call durations in Ncell, and normal distributions appear in NEPSE stock returns.
• Worked examples use small numbers to trace probability calculations step-by-step, linking theory to practical scenarios like loan interest or traffic flow.
• Exam focus: Identify distributions from scenarios, calculate probabilities, and apply stochastic processes to simulate dynamic systems.

Core Concepts: Probability Distributions

1. Discrete vs. Continuous Distributions

Probability distributions describe how likely different outcomes are for a random variable. They are classified into two types:

• Discrete Distributions: For countable outcomes (e.g., number of customers, defects).
  • Binomial Distribution: Models the number of successes in n independent trials (each with success probability p).
    • Parameters: n (trials), p (probability of success).
    • PDF: $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$.
    • Example: Number of successful online orders (e.g., Daraz) out of 10 attempts, where each order succeeds with 80% probability.
    • Worked Example: Problem: A Daraz seller ships 5 orders/day with 90% success rate. What’s the probability of exactly 4 successful shipments? Solution: $$P(X=4)=\left(\genfrac{}{}{0ex}{}{5}{4}\right)(0.9{)}^4(0.1{)}^1=5\times 0.6561\times 0.1=0.32805(\text{or }32.8\%).$$

• Poisson Distribution: Models rare events over time/space (e.g., customer arrivals, defects).

  • Parameters: λ (average rate of events per interval).
  • PDF: $P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$.
  • Example: Customers arriving at an eSewa kiosk (λ = 5/hour). Probability of 3 arrivals in 1 hour: $$P(X=3)=\frac{e^{-5}{5}^3}{3!}=\frac{125}{6e^5}\approx 0.1004(\text{or }10.04\%).$$

• Continuous Distributions: For uncountable outcomes (e.g., time, measurement).

  • Exponential Distribution: Models time between independent events (e.g., call durations, machine failures).

    • Parameters: λ (rate parameter, inverse of mean).
    • PDF: $f(x)=\lambda e^{-\lambda x}$.
    • Example: Ncell call durations average 2 minutes (λ = 1/2). Probability a call lasts >3 minutes: $$P(X>3)=e^{-(1/2)\times 3}=e^{-1.5}\approx 0.2231(\text{or }22.31\%).$$

  • Normal Distribution: Models symmetric, bell-shaped data (e.g., heights, stock returns).

    • Parameters: μ (mean), σ (standard deviation).
    • PDF: $f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$.
    • Example: NEPSE stock returns average 10% with σ=2%. Probability of returns >12%: $$Z=\frac{12-10}{2}=1\Rightarrow P(Z>1)=1-\Phi (1)\approx 0.1587(\text{or }15.87\%).$$

2. Comparative Table of Distributions

Stochastic Processes

Stochastic processes model systems evolving randomly over time. Key types:

flowchart TD
    A["New Customer"] -->|"0.6"| B["Regular"]
    A -->|"0.1"| C["Lost"]
    B -->|"0.7"| B
    B -->|"0.2"| C
    C -->|"0.3"| B
    C -->|"0.7"| C

1. Markov Chains

• Definition: A sequence of states where transitions depend only on the current state (memoryless property).
• Example: Customer loyalty in a bank:
  • States: {New, Regular, Lost}.
  • Transition matrix:

    From\To   | New   | Regular | Lost
    ----------------------------
    New       | 0.3   | 0.6     | 0.1
    Regular   | 0.1   | 0.7     | 0.2
    Lost      | 0.0   | 0.3     | 0.7
    

  • Worked Example: Starting with a "New" customer, probability of being "Lost" after 2 steps: $$P(\text{Lost at step 2})=0.3\times 0.1+0.6\times 0.2+0.1\times 0.7=0.03+0.12+0.07=\mathrm{0.22.}$$
  • Visualization:

2. Renewal Processes

• Definition: Models repeated events where inter-arrival times are independent and identically distributed (i.i.d.).
• Example: Bus arrivals at a Pokhara terminal (exponential inter-arrival times, λ = 1/10 minutes).
  • Worked Example: Probability no bus arrives in 5 minutes: $$P(T>5)=e^{-\lambda \times 5}=e^{-0.5}\approx \mathrm{0.6065.}$$

## In the Real World

1. Daraz Order Processing:

   • Idea: Poisson distribution models order arrivals (λ = 20 orders/hour).
   • How: Simulate peak hours (e.g., 6–9 PM) to predict server load or staffing needs. Example: Probability of >25 orders in 1 hour: $$P(X>25)=1-\sum _{k=0}^{25}\frac{e^{-20}20^k}{k!}\approx 0.0821(\text{or }8.21\%).$$

2. Ncell Customer Service:

   • Idea: Exponential distribution models call durations (λ = 1/3 minutes).
   • How: Estimate queue lengths. Example: Probability a call lasts >4 minutes: $$P(X>4)=e^{-(1/3)\times 4}=e^{-4/3}\approx \mathrm{0.2636.}$$

3. NEPSE Stock Returns:

   • Idea: Normal distribution models daily returns (μ = 0.5%, σ = 1.2%).
   • How: Assess risk. Example: Probability of a >2% return: $$Z=\frac{2-0.5}{1.2}\approx 1.25\Rightarrow P(Z>1.25)\approx \mathrm{0.1056.}$$

4. Khalti Transaction Fraud:

   • Idea: Binomial distribution models fraudulent transactions (p = 0.001 per transaction).
   • Worked Example: Probability of ≥2 frauds in 1000 transactions: $$P(X\ge 2)=1-P(X=0)-P(X=1)=1-(0.999{)}^{1000}-(\genfrac{}{}{0ex}{}{1000}{1})(0.001{)}^1(0.999{)}^{999}\approx \mathrm{0.6321.}$$

## Exam Tip

1. Identify the Right Distribution:

   • Discrete counts? → Binomial/Poisson.
   • Time between events? → Exponential.
   • Symmetric data? → Normal.
   • Example Question: "Customers arrive at a rate of 10/hour. What distribution models arrivals?" Answer: Poisson (λ = 10).

2. Calculate Probabilities Step-by-Step:

   • Show all intermediate steps (e.g., binomial coefficients, Z-scores).
   • Use tables for normal distributions if calculators aren’t allowed.

3. Stochastic Processes:

   • For Markov chains, multiply transition probabilities across steps.
   • For renewal processes, use the CDF of the inter-arrival distribution.

4. Real-World Scenarios:

   • Link theory to examples like:
     • "A bank processes loans with exponential service times (μ = 5 loans/hour). What’s the probability a loan takes >10 minutes?" Solution: $P(X>10)=e^{-5\times (10/60)}=e^{-5/6}\approx 0.4066$.

5. Common Pitfalls:

   • Confusing Poisson (rate λ) with exponential (rate λ for time).
   • Forgetting to convert to Z-scores for normal distributions.
   • Misapplying binomial parameters (e.g., using n as probability).

In the real world

• Daraz Order Processing: Uses Poisson distribution to model order arrivals (λ=20 orders/hour) for predicting peak-hour server load. Example: Probability of >25 orders in 1 hour is 8.21% (used for staffing decisions).
• Ncell Customer Service: Applies exponential distribution (λ=1/3 minutes) to estimate call durations, helping optimize call-center staffing. Example: 60.65% chance no bus arrives in 5 minutes (renewal process for Pokhara terminal).
• NEPSE Stock Returns: Normal distribution (μ=10%, σ=2%) models daily returns, guiding investment strategies. Example: 15.87% chance of returns >12%.