href="#fb3_img_img_ba07927e-168b-57e9-8429-e76efde273ea.png" alt="script upper X"/> can then be the identity function. (It is usual that the values of a random variable are represented as real numbers2 ; with expected—mean or average—value, variance, and so on; which are also real numbers.)
But no matter what way this construction is done, the classical, rigorous mathematical representation by measurable function is evidently more complicated than the naive or natural view of the coin tossing experiment. In contrast, the purpose of this book is to provide a rigorous theory of stochastic integration/summation which (like [MTRV]) bypasses the “measurable function” view, and which is closer to the “naive realistic” view.
Throw a pair of dice and, whenever the sum of the numbers observed exceeds 10, pay out a wager equal to the sum of the two numbers thrown, and otherwise receive a payment equal to the smaller of the two numbers observed. If the two are the same number (with sum not exceeding 10) then the payout is that number.
In Example 3 take sample space
Observation of a throw of the pair of dice can be represented by a listing of the possible joint outcomes
for each
The integral in this case reduces to the sum of a finite number of terms.
The payoff from the wager in Example 3 is a randomly variable amount given by
In this case,
where, again, the Lebesgue integral reduces (trivially) to a finite sum of terms.
There are many alternative ways of representing mathematically the unpredictable payout of this wager, as the following illustration shows. The outcome of the wager is the value
Examining each of the 36 pairs in
For instance, of the 36 possible pairs of throws
