1. Heading: Normal approximations to binomial and to Poisson distributions, chapter 5
Do now: • Which diagram on the right illustrates which distribution?
Which of these distributions are discrete (meaning that the random variable takes only whole-number values) and which are continuous (meaning that it can take any real-number value)?
For the binomial distribution B(N,p), what is the probability of k successes in the N trials? What are the mean and variance of the distribution?
For the Poisson distribution Po(λ), what is the probability of k occurrences? What are the mean and variance of the distribution?
Roughly when can we use the Poisson distribution as an approximation for the binomial distribution? And Poisson distribution for what λ?
The tables in the back of your book - how big are the values of N they go up to for the binomial cumulative distribution, and how big are the values of λ they go up to for the Poisson CD?

2. As n gets large, the binomial distribution becomes more and more like the normal distribution

3. The binomial gets close to the normal quicker when p is around 0
The binomial gets close to the normal quicker when p is around 0.5, but it gets close to the normal eventually, for big enough n, whatever p is. Below is for p=0.25 and n=48

4. Rule of thumb
The rule of thumb is that you can use normal as an approximation for binomial if np>5 and nq>5 too. But Edexcel doesn't ask you to know that.

5. For big enough n, binomial approximates both to Poisson and to normal, so for big enough λ, Poisson also approximates to normal

6. Rules of thumb (which you need not know)
• Poisson approx for binomial – n big and p small, meaning roughly n ≥ 20 and p ≤ 0.05, and np < 10
• Normal approx for binomial – n big and p not too small, meaning roughly np > 5 and nq > 5
• Normal approx to Poisson – λ large, meaning roughly λ > 10

7. The Central Limit Theorem tells us that the total of any large number of independent random variables tends to approximate the normal distribution. Edexcel doesn't ask for knowledge of that, either.
However, the Central Limit Theorem is important. It's often taken as implying that almost every real-life distribution is roughly normal. However, it doesn't imply that. Lots of real-life distributions are "Pareto" (more like Benford's Law). Nassim Nicholas Taleb's book about that fact has been widely read since the crash of 2008.

8. What we'll do Experiment with binomial approximation to normal
Learn about continuity corrections when using the approximation
Practise using the approximation

9. Homework: by Wednesday
Complete Ex. 5D, all
Ex. 5B, Q.1-2
Ex. 5C, Q.1-3