Discrete & Continuous Distributions: Probability Models & Applications

TAKEAWAYS:

• Discrete vs. Continuous: Discrete distributions (e.g., binomial) count outcomes (0, 1, 2...), while continuous distributions (e.g., normal) measure ranges (height, time).
• Probability Mass Function (PMF): For discrete variables, $P(X=x)$ gives the probability of a specific value (e.g., $P(X=2)$ in a binomial trial).
• Probability Density Function (PDF): For continuous variables, $f(x)$ gives the density over an interval (e.g., $P(50<X<60)$ under a normal curve).
• Expected Value (Mean): $E(X)=\sum x\cdot P(X)$ for discrete; $E(X)=\int x\cdot f(x)dx$ for continuous. Critical for risk assessment (e.g., loan defaults).
• Normal Distribution: Symmetric bell curve defined by mean ($\mu$) and standard deviation ($\sigma$). Used in quality control (e.g., NTC’s call-center response times) and finance (e.g., NEPSE stock returns).
• Binomial vs. Poisson: Binomial models fixed trials with success/failure (e.g., Daraz’s 10% order cancellation rate); Poisson models rare events over time/space (e.g., Pathao’s 3 rides/hour demand spikes).

1. Random Variables: The Bridge Between Data and Probability

A random variable (RV) assigns numerical values to outcomes of a random experiment. It connects real-world events to probability theory.

Types of Random Variables

classDiagram
    class RandomVariable {
        <<abstract>>
        +P(X) or f(x)
    }
    class DiscreteRV {
        +Countable outcomes
        +PMF: P(X = x)
        +Example: Binomial, Poisson
    }
    class ContinuousRV {
        +Uncountable outcomes
        +PDF: f(x)
        +Example: Normal, Uniform
    }
    RandomVariable <|-- DiscreteRV
    RandomVariable <|-- ContinuousRV

Key Idea: Discrete RVs use Probability Mass Function (PMF), while continuous RV use Probability Density Function (PDF).

Worked Example 1: Identifying RVs Scenario: A bank processes loan applications. Let $X$ = number of defaults in a month.

• Is $X$ discrete or continuous? Discrete (defaults are counted: 0, 1, 2,...).
• What’s the PMF? $P(X=k)$ = probability of exactly $k$ defaults.

Visual:

2. Discrete Probability Distributions

A. Binomial Distribution

Definition: Models the number of successes ($k$) in $n$ independent trials, each with success probability $p$. PMF: $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k},k=0,1,\dots ,n$$ Mean/Variance: $$E(X)=np,\text{Var}(X)=np(1-p)$$

Real-World Example: Daraz Order Cancellations

• Scenario: Daraz observes 10% of orders are canceled ($p=0.10$). For 15 orders ($n=15$):
  • What’s the probability at least 3 are canceled?
  • Solution: Use binomial PMF for $P(X\ge 3)=1-P(X\le 2)$.

Worked Example 2: Binomial PMF Data: Fit a binomial distribution to $X$ (number of defective chips in a batch of 5), with $p=0.2$:

Steps:

1. Verify $\sum f=64=2^5$ (total trials).
2. Check $p=\frac{\text{successes}}{\text{trials}}=\frac{32+32+16+4+2}{5\times 64}=0.2$.
3. Visual:

Exam Tip: Always check if the data fits binomial assumptions:

• Fixed $n$ trials.
• Independent trials.
• Constant $p$.

B. Poisson Distribution

Definition: Models rare events over time/space (e.g., accidents, calls to NTC). PMF: $$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!},k=0,1,2,\dots$$ Mean/Variance: $$E(X)=\text{Var}(X)=\lambda$$

Real-World Example: Pathao Ride Requests

• Scenario: Pathao receives an average of $\lambda =3$ ride requests/hour. What’s the probability of exactly 5 requests in an hour?
• Solution: $P(X=5)=\frac{e^{-3}{3}^5}{5!}\approx 0.1008$.

Worked Example 3: Poisson vs. Binomial Scenario: A call center receives 20 calls/day ($n=20$, $p=0.1$ for a "complaint" call).

• Binomial: $P(X=3)=\left(\genfrac{}{}{0ex}{}{20}{3}\right)(0.1{)}^3(0.9{)}^{17}\approx 0.233$.
• Poisson Approx: $\lambda =np=2$. $P(X=3)=\frac{e^{-2}{2}^3}{3!}\approx 0.180$.

Visual Comparison:

3. Continuous Probability Distributions

A. Uniform Distribution

Definition: All outcomes in $[a,b]$ are equally likely. PDF: $$f(x)=\{\begin{array}{ll}\frac{1}{b-a},& a\le x\le b\\ 0,& \text{elsewhere}\end{array}$$ Mean/Variance: $$E(X)=\frac{a+b}{2},\text{Var}(X)=\frac{(b-a{)}^2}{12}$$

Real-World Example: NTC Call Wait Times

• Scenario: NTC guarantees call wait times are uniformly distributed between 1 and 5 minutes.
  • Probability a call waits >3 minutes: $P(X>3)=\frac{5-3}{5-1}=0.5$.

Worked Example 4: Uniform CDF Scenario: A bus arrives at a stop every 10 minutes. If arrival times are uniform:

• Find $P(2<X<7)$ where $X$ = wait time in minutes.
• Solution: $$P(2<X<7)=\frac{7-2}{10-0}=0.5$$
• Visual:

B. Exponential Distribution

Definition: Models time between events (e.g., failures, arrivals). PDF: $$f(x)=\{\begin{array}{ll}\lambda e^{-\lambda x},& x\ge 0\\ 0,& \text{elsewhere}\end{array}$$ Mean/Variance: $$E(X)=\frac{1}{\lambda },\text{Var}(X)=\frac{1}{{\lambda }^2}$$

Real-World Example: Khalti Transaction Failures

• Scenario: Failures occur at rate $\lambda =0.5$ per hour.
  • Probability a transaction fails within 1 hour: $P(X\le 1)=1-e^{-0.5\times 1}\approx 0.393$.

Worked Example 5: Exponential CDF Scenario: A computer’s hard drive fails at rate $\lambda =0.2$ failures/year.

• Find $P(X>5)$ (drive lasts >5 years).
• Solution: $$P(X>5)=e^{-0.2\times 5}=e^{-1}\approx 0.368$$
• Visual:

C. Normal Distribution

Definition: Symmetric bell curve defined by $\mu$ (mean) and $\sigma$ (standard deviation). PDF: $$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$ Standard Normal: $\mu =0$, $\sigma =1$. Use $Z$-scores: $$Z=\frac{X-\mu }{\sigma }$$

Real-World Example: NEPSE Stock Returns

• Scenario: Daily returns are $N(\mu =0.5\%,\sigma =1.2\%)$.
  • Probability return >1%: $P(X>1)=P(Z>\frac{1-0.5}{1.2})=P(Z>0.417)\approx 0.338$.

Worked Example 6: Normal CDF Scenario: Exam scores are $N(70,10)$. Find $P(60<X<80)$.

• Steps:
  1. Standardize: $Z_1=\frac{60-70}{10}=-1$, $Z_2=\frac{80-70}{10}=1$.
  2. Use $Z$-table: $P(-1<Z<1)=0.6826$.
• Visual:

Exam Tip: For normal distributions:

• Always standardize ($Z$-score) before using tables.
• Remember empirical rule: 68% within $\mu \pm \sigma$, 95% within $\mu \pm 2\sigma$.

4. Comparing Discrete and Continuous Distributions

5. Applications in Nepal’s Tech and Finance

A. Quality Control (NTC, Daraz)

• Binomial: Daraz checks 10% of orders for defects. If 5% are defective, what’s the probability a sample of 20 has exactly 2 defects? $$P(X=2)=\left(\genfrac{}{}{0ex}{}{20}{2}\right)(0.05{)}^2(0.95{)}^{18}\approx 0.283$$
• Normal: NTC monitors call durations. If $\mu =3$ mins, $\sigma =0.5$ mins, what’s the probability a call lasts >4 mins? $$P(X>4)=P(Z>\frac{4-3}{0.5})=P(Z>2)\approx 0.0228$$

B. Finance (Banks, NEPSE)

• Poisson: A bank processes 5 loan applications/hour. Probability of 0 applications in 30 mins ($\lambda =2.5$): $$P(X=0)=e^{-2.5}\approx 0.082$$
• Exponential: NEPSE stock crashes occur at rate $\lambda =0.1$ per month. Probability of no crash in 6 months: $$P(X>6)=e^{-0.1\times 6}=e^{-0.6}\approx 0.549$$

C. Traffic and Logistics (Pathao, Kathmandu Traffic)

• Uniform: Traffic lights cycle every 60 seconds. If arrival times are uniform, probability a car waits >45 seconds: $$P(X>45)=\frac{60-45}{60}=0.25$$
• Normal: Pathao delivery times are $N(45,5)$ mins. Probability a delivery takes >50 mins: $$P(X>50)=P(Z>\frac{50-45}{5})=P(Z>1)\approx 0.1587$$

6. Solving Exam-Style Problems

Problem 1: Binomial Distribution (5 Marks)

A virus enters a system via email (30%) or internet (40%). Independent events.

1. Probability of no infection: $P(\text{neither})=(1-0.3)(1-0.4)=0.42$.
2. Probability of both: $P(\text{both})=0.3\times 0.4=0.12$.
3. Visual:

pie
    title Virus Entry Paths
    "Email Only": 0.30
    "Internet Only": 0.40
    "Both": 0.12
    "Neither": 0.42

Problem 2: Normal Distribution (5 Marks)

Exam data: 10% scored <20, 95% scored <75. Assume normal.

1. Find $\mu$ and $\sigma$:
   • $P(X<20)=0.10\Rightarrow Z=-1.28\Rightarrow \mu -1.28\sigma =20$.
   • $P(X<75)=0.95\Rightarrow Z=1.645\Rightarrow \mu +1.645\sigma =75$.
2. Solve: $$\mu =\frac{20+75}{2.925}\approx 35.9,\sigma \approx 11.6$$
3. Visual:

Problem 3: PDF and CDF (5 Marks)

Given: $$f(x)=\{\begin{array}{ll}k(1-x{)}^2,& 0<x<1\\ 0,& \text{elsewhere}\end{array}$$

1. Find $k$: $${\int }_0^{1}k(1-x{)}^{2}dx=1\Rightarrow k=3$$
2. Find $P(0.5<X<0.8)$: $${\int }_{0.5}^{0.8}3(1-x{)}^{2}dx={\left[-(1-x{)}^3\right]}_{0.5}^{0.8}=0.189$$
3. Visual:

## In the Real World

1. eSewa and Khalti (Payment Failures)

   • Poisson Distribution: Models rare transaction failures. If Khalti processes 1000 transactions/day with $\lambda =2$ failures/day, the probability of no failures in a day is $e^{-2}\approx 0.135$.

2. Daraz and NTC (Quality Control)

   • Binomial Distribution: Daraz samples 5% of orders for quality checks. If 2% are defective, the probability a batch of 20 has exactly 1 defect is $\left(\genfrac{}{}{0ex}{}{20}{1}\right)(0.02{)}^1(0.98{)}^{19}\approx 0.277$.

3. Pathao and Ncell (Customer Wait Times)

   • Exponential Distribution: Ncell’s customer service wait times follow $\lambda =0.2$ calls/min. The probability a customer waits >5 minutes is $e^{-0.2\times 5}=e^{-1}\approx 0.368$.

4. NEPSE and Banks (Risk Assessment)

   • Normal Distribution: NEPSE stock returns are modeled as $N(0.5\%,1.2\%)$. Investors use this to calculate the probability of returns exceeding 2%: $$P(X>2)=P\left(Z>\frac{2-0.5}{1.2}\right)=P(Z>1.25)\approx \mathrm{0.106.}$$

## Exam Tip

1. Identify the Distribution:

   • Countable data? → Discrete (Binomial/Poisson).
   • Measurable data? → Continuous (Normal/Uniform).
   • Rare events over time? → Poisson.
   • Fixed trials with success/failure? → Binomial.

2. Check Assumptions:

   • Binomial: Fixed $n$, independent trials, constant $p$.
   • Normal: Symmetric, bell-shaped data (use mean/median ≈ mode).

3. Visualize:

   • Always sketch the distribution (bar chart for discrete, curve for continuous).
   • Shade the area corresponding to the probability asked.

4. Standardize for Normal:

   • Convert $X$ to $Z$ using $Z=\frac{X-\mu }{\sigma }$.
   • Use $Z$-tables or calculator for probabilities.

5. Common Pitfalls:

   • Discrete vs. Continuous: $P(X=x)$ is valid for discrete; for continuous, use $P(a<X<b)$.
   • Binomial vs. Poisson: Poisson approximates binomial when $n$ is large and $p$ is small ($np<5$).

6. Worked Example Trace:

   • For every problem, write:
     1. Distribution identified.
     2. Parameters ($\mu$, $\sigma$, $n$, $p$, $\lambda$).
     3. Formula applied.
     4. Calculation steps.
     5. Final probability with units (e.g., "0.234 or 23.4%").

Final Note: Master the shape of each distribution (binomial’s skew, normal’s symmetry, exponential’s decay) and how parameters shift them. In exams, always justify your choice of distribution before calculations.