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How to Value Your Rich Elderly Aunt

Philip Cooper, 26 November 2002, 10 am – 3.30 pm

Task 3

[Note to teachers: again, there may not be time to do everything. You could let groups of students do one or more of these integrations, rather than have everyone start at the beginning. Year 12 students, who have not yet met integration, could try using the Trapezium Rule, or Hardy's 39a formula with the simple example given. Graphic calculators could be used to confirm the results. Once students have had a go at the methods, get them to focus on the original problem, so they use the techniques to actually get an answer to it. Again, speakers need to be ready when you rejoin the videoconference to report back on how you got on.]

3.1 Try out the special case of the Trapezium Rule (see Additional Notes) for integer end points, and spacings of length 1, by finding the value of

\begin{displaymath}\int_{x=0}^{x=10} \par (x^{2} + 2x + 1)dx \end{displaymath}

using the values of the function at the points x = {1, 2, 3, ..., 10}.

3.2 This is a straightforward function to integrate exactly so you will be able to check the accuracy of the Trapezium Rule in this case. What is the % error? (Use a graphic calculator to find the value of the definite integral if you have not yet met numerical integration).

3.3 Try out Hardy’s 39a formula, by investigating the following definite integral

\begin{displaymath}\int_{x=0}^{x=\infty} \par\frac{2x}{(1 + x^{2})^{2}} dx \end{displaymath}

You will first need to plot the function to check its shape (remember, it has to be skewed to the right - have a right hand tail that is fatter than the left hand tail) and have a single ‘hump’ towards the left hand end of the possible range of values. The function starts at the origin and the time axis is an asymptote to the curve for large values of t(as $ t \rightarrow \infty $ so $ f(t) \rightarrow 0 $ ).
Then you will need to determine a suitable value for n where $ f(7n)\cong 0 $ .

Hardy’s 39a formula is:

\begin{displaymath}\int_{t=0}^{t=\infty} \par f(t) dt = n[0.28f(0) + 1.62f(n) + 2.20f(3n) + 1.62f(5n) \par + 0.56f(6n) + 1.62f(7n)] \end{displaymath}

Then you can find the six values of the function that are necessary to use Hardy’s formula; use Hardy’s ‘magic’ coefficients, don’t forget to multiply by n and you will have a numerical value for the integral.

3.4 You can use your knowledge of integration by finding anti-derivatives to find the accurate value and compare this value to the one you obtained using Hardy’s formula. Alternatively you could use a graphic calculator to find the value of the definite integral. What is the % error?

3.5 You may have already learnt about the exponential function, if so you can try Hardy’s 39a formula (as in 3.3) to find the value of

\begin{displaymath}\int_{x=0}^{x=\infty} \par x e^{-x^{2}} dx \end{displaymath}

3.6 Check the value you obtained for this using Hardy’s 39a formula by using direct integration, or using a graphic calculator.

3.7 If you have tried out Hardy’s 39a formula on a simple function, you are ready to use it to find the probability that you outlive Auntie. Use the graph and table below to work it out.

Can you use the probability to work out a fair value for you to sell your interest in the £10,000 Trust? What have we not taken into account yet? How do you think these factors would change things?

Data for producing graph: total area under curve gives the probability of Auntie dying first

t  
males
$ \par\ell_{20+t} $  
females
$ \par\ell_{60+t} $  
females
$ \par\mu_{60+t} $  
1#1  
0
98 496
91 732
0.007 86
0.007 86
7
97 900
94 567
0.016 09
0.014 80
14
97 273
72 048
0.032 03
0.024 84
21
96 334
52 293
0.064 64
0.036 04
28
94 668
27 017
0.131 74
0.037 29
35
92 217
7 206
0.257 32
0.018 72
42
84 173
663
0.437 69
0.002 70
49
70 598
12
0.731 91
0.000 07

 

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© 2002 Millennium Mathematics Project, University of Cambridge