Collections of past-paper questions by topic
- Sampling and hypothesis testing
- Continuous random variables
- Distributions: binomial, Poisson, normal, uniform
Collections of full past papers by date
- S2 papers 2017 to 2010, with mark scheme for 2017 paper but not others
- Mark scheme for June 2014 paper
- Mark scheme for June 2014 (R) paper
- Worked answers for January 2013 S2 past paper
- S2 papers 2014 to 2010
- S2 papers 2009 to 2007
Mark schemes for other S2 past papers can be found at the Physics and Maths tutor website
Week 1: How many throws do you need to be confident that a biased dice is biased?
Or: how weird does the result have to be to make you confident that the result is not random?
What does “random” mean for a fair dice?
The binomial distribution; coin tosses and dice throws.
- Pascal’s triangle
- Calculating probabilities using nCr
- Calculating probabilities using n!/(n-r)!r!
- Calculating probabilities using tables
When to use the binomial distribution
Starter: The binomial distribution describes numbers of heads (or tails) when you toss a coin, or the number of ones (or sixes, or whatever) when you toss a dice.
Write on the whiteboard some other random variables described by a binomial distribution
Write on the whiteboard what the rules are for a random variable to be described by the binomial distribution.
When we’re sure we have it correct and clear, you will write those rules in your notebook.
Review first problem on the “four problems” sheet
Mean and variance of the binomial distribution
Mean = np and variance = np(1−p)
Standard deviation = √variance = √[np(1−p)]
For the exam, you need only read these formulas from the formula book. But this is how to get them.
If X is a random variable described by the binomial distribution for n trials and probability of success p, which we call B(n,p):
p(X=r) = nCr.pr(1–p)n-r = [n!/(n–r)!r!]. pr(1–p)n-r
The mean of X = E(X) = Σr=0n r.p(X=r)
Do some algebra to calculate E(X)
The variance of X = E(X2)–[E(X)]2
= Σr=0n r2.p(X=r)–mean2
= Σr=0n (r2–r).p(X=r)+ Σr=0n r.p(X=r)
= Σr=0n r(r–1).p(X=r)+ Σr=0n r.p(X=r)
Do some algebra to calculate E(X2), and then calculate variance(X)
S2: Binomial distribution
Ex.1E, Q.4, Q.7
Optional extra: Review exercises, p.64 Q.4, Q.8
Get good with the binomial distribution
Starter activity: Shut your notebook and textbook and write on the whiteboard:
1. when a random variable follows the binomial distribution
2. what B(n,p) means
3. what the formula is for the probability of r successes in n trials when the probability of success in each trial is p
4. what the mean of B(n,p) is
5. what the variance of B(n,p) is
6. what equation connects standard deviation and variance.
Ex.1E Q.8,9. Review exercises p.64 Q.1, Q.2 without using an approximation.
We want to find the probability of a team scoring 3 goals in a football match of 90 minutes. As an approximation, divide the 90 minutes into 30 units of 3 minutes each and take the probability of a goal in each 3-minute unit to be 1/30.
We will then work through an introduction to, and activities about, the Poisson distribution.
Homework for next week: Binomial and Poisson distributions. Review exercises p.64 Q.1,2 , Ex.1E Q.9, Ex.2E Q.1-4.
Getting good with the Poisson distribution, and Poisson approximation to binomial
Starter: Shut your textbook and your notebook, and write on the whiteboard what variables other than goals in football matches might follow a Poisson distribution
We will do Ex.2F Q.2 as a worked example in class.
Then work on Ex.2E Q.1-5.
Then we will summarise the Poisson distribution:
- A random variable X follows a Poisson distribution if it is a count of the number of successes where:
- Successes occur randomly in continuous time or space (i.e. not in a discrete series of trials)
- Successes are independent of each other, and occur at a constant rate over time or space
- Formula: probability of r successes = [λr/r!].e−λ
- Mean=λ. Variance=λ. Standard deviation=√λ
- Poisson can be used as a simple approximation to binomial when n is big and p is small. There is no exact rule for what “big” and “small” mean here, and Edexcel does not demand one. But a good rule of thumb is n≥20 and p≤0.05.
Getting good with the Poisson distribution, and Poisson approximation to binomial
Starter: Shut your textbook and your notebook, and write on the whiteboard the conditions for a random variable to follow the binomial distribution, and the conditions for it to follow the Poisson distribution.
Then we will work on Ex.2F Q.1-10.
The normal distribution is defined by the probability distribution function
The indefinite integral of the function cannot be expressed in terms of functions we already know, but we can calculate the definite integral
and thus see where the factor comes from. It’s a neat example of the occasional pattern in maths where we can solve a problem, paradoxically, by translating into a more complicated problem.
First square the integral, and you get an expression which equals the volume of the shape got by revolving a bell-type curve, round the y-axis.
That volume must be the same as it would be if calculated in polar coordinates. In polar coordinates the infinitesimal unit of area in the base of the shape is rather than , and the height of the shape above that infinitesimal unit is . So:
The variance V of the normal distribution:
By integration by parts:
Go over normal distribution from S1 again
Visual: binomial approximation to normal. (The proof that binomial is approximated by normal for large n is longer than the proof that binomial is approximated by Poisson for large n and small p, but rests on the same basic use of Maclaurin series for ln(1+x). https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem)
Central Limit Theorem: why the normal distribution is so widely used, and the pitfalls
Homework: Ex. 5B Q.1-4
- Discrete and continuous, and which distribution approximates which?
- Which distribution when? Binomial, Poisson, and normal
Problem 3 on our “Four Problems” sheet
Continuous uniform distribution
Homework: Continuous uniform distribution: Ex.4C Q.1, 2, 3
Week 6: Words to use about sampling. Homework: Review Exercise, p.127, Q.6, and p.128 Q.14, page 129 Q.17. Exam practice paper p.131, Q.1
Week 7: no lessons (half-term)
Week 8: Basics of hypothesis testing. Problem 4 on our “Four Problems” sheet. Null hypothesis, alternative hypothesis, significance level. Homework: Ex.7A, Q.1-7.
The basic ideas are explained better than in the textbook, I think, in Moore and Notz’s “Statistics: Concepts and Controversies”: 171222moore-hyp-testing. Moore uses Ha instead of H1 for “alternative hypothesis”.
Week 9: More on hypothesis testing. Critical values, critical region, actual significance level. Ex.7B Q.1-3, 8-10, 11-12.
Week 10: Continuous random variables. Continuous random variables (pdf)
Week 11: no lessons (mock exam week). Homework: work through mock exam paper and mark scheme.
Week 12: “How random can you get?” survey. Review mock exam. Homework: an S2 paper.
Week 13: “How random can you get?” survey. Homework: an S2 paper
Week 14: Another mock exam paper. Revision sheets: Binomial, Poisson, normal: what you have to remember; and Revision sheet for the whole course. S2 paper in class, and S2 test paper in exam conditions.
Week 15: no lessons (end of term on Thursday)
- Four problems which define what you will learn on the course
- Introduction to, and activities about, the Poisson distribution
- Worksheet on discrete and continuous, and Poisson and normal approximations to binomial. This uses these websites http://www.waldomaths.com/PoissBin1L.jsp: binomial approximation to Poisson – visual demonstration; http://www.waldomaths.com/Binomial1L.jsp: binomial approximation to normal – visual demonstration.
- Central Limit Theorem – visual demonstration
- The Central Limit Theorem and the normal distribution
- Exam questions on the continuous uniform distribution
- Introduction to Benford’s Law
- The maths of Benford’s Law
- “How random can you get?” survey
- Do costlier choc chip cookies have more choc chips? Debrief
- How not to do hypothesis testing: the “21 grams” experiment
- Briefing sheet/ worksheet on definitions and explanations required for S2
- Website to help you learn the definitions you need for the exam
- Exam questions on the binomial distribution
- Proof of E and var for Poisson distribution, and that binomial probabilities approach Poisson probabilities as n gets larger with np fixed.
- Analysis of Premier League goals 2013-4
- Other hypothesis testing projects
- Binomial, Poisson, normal: what you have to remember
- Revision sheet for the whole course