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## Derive Session

I introduce a utility function Y_(f,x,), the purpose of which is to compute the value of the function at . In many utility files that come along with the software the trick to achieve this is to compute the limit of the expression at . However, I prefer to use the ITERATE command in the following way:

#1:  Y_(f,x,alpha) := ITERATE(f,x,alpha,1)
The utility function can be used to compute the convolution product of two functions f and g:

#2:  CONV(f,g,x):=INT(Y_(f,x,t)*Y_(g,x,x-t),t,-inf,inf)
The probability density function of adding f n times can be recursively computed in Derive by the following procedure.
#3:  DEN_SUM(n,f,x):=IF(n=1,f,CONV(DEN_SUM(n-1,f,x),f,x))
The standard uniform distribution is available in Derive.
#4:  U(x):=CHI(0,x,1)

Below you see a screen dump in which I compute DEN_SUM(n,U(x),x) for n=1,2,3.

Figure 7.1: for n=1,2,3

In Figure 7.2 you see the results for adding 4 and 5 times. The best result is obtained after applying the Factor command once or twice on the intermediate reult of simplifying DEN_SUM(4,U(x),x) and DEN_SUM(5,U(x),x).

Figure 7.2: for n=4,5

You may recognize Pascal's triangle and conjecture the same general formula as I do:

In Derive I author this formula as the function UN(n,x):

#18: UN(n,x):=1/(2(n-1)!)*SUM((-1)^k*COM(n,k)*SIGN(x-k)*(x-k)^(n-1),k,0,n)

Now the proof by induction can start. First the check that UN(1,x)=U(x).

#19: U(x) - UN(1,x) = 0
Supposing that the formula holds for n I can compute
#20: CONV( UN(n,x), U(x) )
The result of simplification is partly shown on line #21 in Figure 7.3.

Figure 7.3: Proof by induction

What are in short the steps I take above? Well, Fiddling with the F3 key and with manual changes in the Author and Build command lines the intermediate result is rewritten as the formula in line #22. The steps are omitted but I combine integrals and sums, and I interchange summation and integration. In Figure 7.4 below you see that the integration interval is restricted to on the basis of elementary properties of the SIGN function. Derive can compute the integral and with a bit of help in the handling of sums --- intermediate results are in line #25 and #26 --- I find the formula of line #27:

I can verify with Derive that this is indeed equal to UN(n+1,x):

Figure 7.4: Proof by induction (continued)

Figure 7.5: Proof by induction (continued)

Next: Letters about Wester's Up: Talking about Chance Previous: Birth of a

Andre Heck
Sun Apr 23 10:32:10 MDT 1995