I introduce a utility function
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
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.
Below you see a screen dump in which I compute
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
command once or twice on the intermediate reult of simplifying
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
Now the proof by induction can start. First the check that
#19: U(x) - UN(1,x) = 0Supposing that the formula holds for
nI 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
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
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
Figure 7.4: Proof by induction (continued)
Figure 7.5: Proof by induction (continued)