doi: : 10.2478/v10127-012-0025



In life insurance mathematics, the concept of a survival function is commonly
used in life expectancy calculations. The survival function of a random variable X
is defined at x as the probability that X is greater than a specific value x. For
a non-negative random variable whose expected value exists, the expected value
equals the integral of the survival function. We propose to designate this result
as the Darth Vader Rule1. It holds for any type of random variable, although its
most general form relies on the integration by parts formula for the Lebesgue-
-Stieltjes integral, fully developed by H e w i t t [3]. This result, while known (and
stated in F e l l e r [1]), is not widely disseminated except in life insurance mathematics texts; but it is worth knowing and popularizing because it provides an
efficient tool for calculation of expected value, and gives insight into a property
common to all types of random variables.
We give a proof of the Darth Vader Rule which works for all random variables which are non-negative almost surely and whose expected value exists.
The proof is based not on the Lebesgue integral formulation of [3], but on the
generalized Riemann integration of H e n s t o c k and K u r z w e i l [2], [4]. Since
every Lebesgue integrable function is also generalized Riemann integrable, the
proof here includes all cases covered by [3].
While the result is simple to state and comprehend, its proof using Lebesgue
integral theory is somewhat complex.

Author(s): Pat Muldowney — Krzysztof Ostaszewski — Wojciech Wojdowski

Publication Date: 2012

Publication Site: Tatra Mountains Mathematical Publications





During the early part of the eighteenth century, the French government issued a series of bonds to help raise money.  With the decline of the French economy in the 1720s, they were forced to cut the interest rates on the bonds, which drastically diminished the market value of said bonds.  This resulted in the French government having considerable difficulty in raising money via new bond sales.

One Michel Robert Le Pelletier-Desforts, Deputy Finance Minister for France, had a “brilliant” idea as to how to raise the value of existing bonds, encourage the sale of new bonds, and earn some money for the government- a trifecta. His idea was to allow bond owners to buy a lottery ticket linked to the value of their bonds (each ticket costing 1/1000th of the bond’s value). The winner would get the face value of their bond, which was more than what they could get on the market at this point, plus a ‘jackpot’ of 500,000 livres, which would make the winner instantly wealthy.


Unfortunately for the government, and fortunate for those of you who enjoy Voltaire’s work, the mathematics behind this new government fundraising scheme was fundamentally flawed.  You see, if you owned a bond worth a relatively small amount, with the lotto ticket for the bond costing just 1/1000th of the value, you could buy the lotto tickets cheaply, yet your lotto ticket had just as much of a chance of winning as someone who owned a bond for, say, 100,000 livres and had to buy their ticket for 100 livres.

Thus, when La Condamine crunched the numbers, he realized that if he was able to buy up a certain percentage of the existing small bonds, he could then acquire the necessary entrees in the lotto to reasonably ensure he’d win, all while spending significantly less than the jackpot and also making a profit on the bonds themselves when he ultimately won and the government had to pay face value for them.

Author(s): Andy Williamson

Publication Date: 16 May 2013

Publication Site: Today I Found Out