Report Writing

A report should be self-explanatory. It should be capable of being read and understood without reference to the original project description. Thus, for each question, it should contain all of the following:

(a)a statement of the problem;

(b) a full and careful description of how it is investigated;

(c) All relevant results, including graphical and numerical analyses; variables should be carefully defined, and figures and tables should be properly labelled, described and referenced;

(d) relevant analysis, discussion, and conclusions.

It should be written in the third person.

NOT: I think the Central Limit Theorem is true for this example because I see that the graph is normal.

INSTEAD: It can be clearly seen that the graph displays a normal distribution confirming that the Central Limit Theorem holds.

The Central Limit

Theorem

Let X1, X2………. Xn be independent identically distributed random variables with mean µ and variance σ 2.

Let S = X1,+ X2+ ………. +Xn

Then elementary probability theory tells us that E(S) = nµ and var(S) = nσ 2 .

The Central Limit Theorem (CLT) further states that, provided n is not too small, S has an approximately normal distribution with the above mean nµ, and variance nσ 2.

In other words,

S approx ~ N(nµ, nσ 2)

The approximation improves as n increases.

We will use R to demonstrate the CLT.

Let X1,X2……X6 come from the Uniform distribution, U(0,1)

0 1

1

For any uniform distribution on [A,B],

µ is equal to

and variance, σ2, is equal to

2

A B

2( )

12

B A

So for our distribution, µ= 1/2 and

σ2 = 1/12

The Central Limit Theorem therefore states that S should have an approximately normal distribution with mean nµ (i.e. 6 x 0.5 = 3)

and var nσ2 (i.e. 6 x 1/12 = 0.5)

This gives standard deviation 0.7071

In other words,

S approx ~ N(3, 0.70712)

Generate 10 000 results in each of six vectors for the uniform distribution on [0,1] in R.

> x1=runif(10000)> x2=runif(10000)> x3=runif(10000)> x4=runif(10000)> x5=runif(10000)> x6=runif(10000)>

Let S = X1,+ X2+ ………. +X6

> s=x1+x2+x3+x4+x5+x6> hist(s,nclass=20)>

Consider the mean and standard deviation of S

> mean(s)[1] 3.002503> sd(s)[1] 0.7070773>

This agrees with our earlier calculations

A method of examining whether the distribution is approximately normal is by producing a normal Q-Q plot.

This is a plot of the sorted values of the vector S (the “data”) against what is in effect a idealised sample of the same size from the N(0,1) distribution.

If the CLT holds good, i.e. if S is approximately normal, then the plot should show an approximate straight line with intercept equal to the mean of S (here 3) and slope equal to the standard deviation of S (here 0.707).

> qqnorm(s)>

> qqnorm(s)>

> qqnorm(s)>

4.4 – 1.8

4

= 0.7 to 1 DP

From these plots it seems that agreement with the normal distribution is very good, despite the fact that we have only taken n = 6, i.e. the convergence is very rapid!