FAD1015 Tutorial 8 — Uniform & Exponential Distributions

Tutorial questions on continuous uniform and exponential distributions. Source file: FAD1015 25-26 Tutorial 8 Questions.pdf

Summary

Problem set covering uniform distribution (constant probability) and exponential distribution (waiting times, memoryless property) with practical applications.

Topics Covered

1. Uniform Distribution

• PDF and CDF calculations
• Expected value and variance
• Probability over intervals

2. Exponential Distribution

• PDF: f(x) = λe^(-λx)
• CDF: F(x) = 1 - e^(-λx)
• Mean and standard deviation: 1/λ
• Memoryless property applications

3. Relationship to Poisson

• Poisson: events per unit time
• Exponential: time between events
• λ parameter connection

Key Formulas

Uniform Distribution [a, b]:

• PDF: f(x) = 1/(b-a)
• E[X] = (a + b)/2
• Var(X) = (b-a)²/12

Exponential Distribution:

• PDF: f(x) = λe^(-λx), x ≥ 0
• CDF: F(x) = 1 - e^(-λx)
• E[X] = 1/λ
• Var(X) = 1/λ²

Memoryless Property: $$P(X > s + t \mid X > s) = P(X > t) = e^{-\lambda t}$$

Problem Types

1. Uniform: Finding probabilities over intervals, expected values
2. Exponential: Waiting time problems, reliability, survival analysis
3. Memoryless: Conditional probability problems
4. Poisson-Exponential Link: Converting between event counts and waiting times

Related Lectures

• FAD1015 L17-L18 — Uniform & Exponential Distributions + R Intro
• FAD1015 L14 — Poisson Distribution — related distribution
• FAD1015 L15-L16 — Normal Distribution & Approximation — other continuous distributions

Related Concepts

• Probability Distributions

Related Course Page

• FAD1015 - Mathematics III