| > (a) In a box we put two marbles, one black and one white. We choose
> one marble at random. If it is white, we put it back in the box, add
> an extra white marble to the box, shake the box, and draw again,
> continuing to replace the marble along with an extra white marble every
> time a white marble is chosen, until the black marble is chosen and the
> game ends. What is the average number of marbles chosen?
Working out the probabilities yields 1/2 + 1/3 + 1/4 + 1/5 + ...,
which diverges, so the average number of marbles chosen doesn't exist!
(b) What is the average number of marbles chosen if we add an extra
white marble only after every _second_ white marble that is chosen?
This appears to be: Sum(n=1 to inf)((4n� + n - 1)/(n� * (n+1)�))
Which converges, at least...
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| That .1:
> This appears to be: Sum(n=1 to inf)((4n� + n - 1)/(n� * (n+1)�))
Maple says that converges to 1 + pi^2/6, but how do you get that
formula? I figure there's a 1/2 chance of 1 marble, 1/2^2 of 2,
1/2^2 * 1/3 of 3, 1/2^2 * 1/3^2 of 4, . . .
If p(i) is the product from j=2 to i of 1/j^2, then the average number
of marbles chosen is the sum from i=1 to infinity of
[ (2i-1)/(i+1) + (2i)/(i+1)^2 ] * p(i),
where I have grouped the numbers chosen into pairs -- 2i-1 and 2i are
numbers 1 and 2 then 3 and 4, et cetera. This sums, numerically, to
about 1.409, which is different from 1 + pi^2/6, which is about 2.645.
-- edp
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| >> This appears to be: Sum(n=1 to inf)((4n� + n - 1)/(n� * (n+1)�))
>Maple says that converges to 1 + pi^2/6, but how do you get that
>formula?
If you expand the summand in partial fractions you will get
something of the form
a/n + b/(n+1) + c/n^2 + d/(n+1)^2
I didn't work it out, but the first two terms probably have
cancellation leaving well known sums in terms of 1/n^2 (or more
generally polylogarithms or digamma functions if I recall, which
is how Maple knows how to do it.)
From an EE point of view, the terms of the summand are the Fourier
coefficients of some periodic waveform. Parsevals relation relates
these to the RMS value of the waveform itself, which is going to be
some simple thing involving linear and quadratic terms in this case.
Which has little to do with formulating the origional problem but
I just had to put my two cents in :-)
- Jim [too busy with work to be reading notes]
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