Disclaimer: This post is written based on the syllabus for Cambridge International AS and A Levels, and may not be applicable for higher level studies.
Approximation
- Binomial to Normal:
- Conditions*: $np > 5$ and $nq > 5$ ($n$ is sufficiently large).
- Apply continuity correction.
- Binomial to Poisson:
- A Poisson distribution can be used to model a discrete probability distribution in which the events
- occur singly,
- at random and independently,
- in a given interval of space or time.
- The mean and variance of a Poisson distribution are equal.
- Conditions*: $n > 50$ and $np < 5$ ($n$ is large, $p$ is small/rare event).
- Poisson to Normal:
- Conditions*: $\lambda > 15$
- Apply continuity correction.
Central Limit Theorem
The central limit theorem (CLT) states that, provided $n$ is large, the distribution of sample means of size $n$ is:
\(\overline{X}(n) \sim N\left(\mu, \frac{\sigma^2}{n}\right),\)where the original population has mean $\mu$ and variance $\sigma^2$.
- CLT can be used for sample size $n > 50$* (or $n > 30$**).
*Rule of thumb
**For Further Statistics
References:
- Chalmers, Dean. Cambridge International AS & A Level Mathematics: Probability & Statistics 1 - Coursebook. Cambridge University Press, 2018.
- Kranat, Jayne. Cambridge International AS & A Level Mathematics: Probability & Statistics 2 - Coursebook. Cambridge University Press, 2018.
- McKevley, Lee, and Crozier, Martin. Cambridge International AS & A Level Further Mathematics - Coursebook. Cambridge University Press, 2018.