Study Notes
Approximations in statistics involve using one type of distribution to estimate another when certain conditions are met. This helps simplify calculations and solve real-life problems.
- Poisson Approximation — Used when a binomial distribution has a large number of trials and a small probability of success. Example: If X ~ B(n, p) with n ≥ 50 and np ≤ 5, then X can be approximated by Po(np).
- Normal Approximation to Binomial — Used when a binomial distribution has a large number of trials and both np and np(1-p) are greater than 5. Example: X ~ B(n, p) can be approximated by N(np, np(1-p)) with continuity correction.
- Normal Approximation to Poisson — Used when the mean of a Poisson distribution is greater than 15. Example: Po(λ) can be approximated by N(λ, λ) with continuity correction.
Exam Tips
Key Definitions to Remember
- Poisson Approximation: Used for binomial distributions with large n and small p.
- Normal Approximation to Binomial: Used when np > 5 and np(1-p) > 5.
- Normal Approximation to Poisson: Used when λ > 15.
Common Confusions
- Forgetting to apply continuity correction when using normal approximations.
- Misidentifying when to use Poisson vs. normal approximations.
Typical Exam Questions
- When can a binomial distribution be approximated by a Poisson distribution? When n ≥ 50 and np ≤ 5.
- How do you apply a continuity correction? Adjust discrete values to continuous intervals, e.g., X = 13 becomes 12.5 ≤ X ≤ 13.5.
- What conditions allow a Poisson distribution to be approximated by a normal distribution? When λ > 15.
What Examiners Usually Test
- Understanding of when and how to apply different approximations.
- Ability to correctly apply continuity corrections.
- Calculating probabilities using approximations.