3.3. DURATION OF THE GAMBLER’S RUIN 107
in the gambler’s ruin, just as for the probability of ruin or victory.Vocabulary
- Expectation by conditioningis the process of deriving an expec-
tation by conditioning the outcome over an exhaustive, mutually ex-
clusive set of events, each of which leads to a simpler probability cal-
culation, then weighting by the probability of each outcome of the
conditioning events. - First Step Analysisis how J. Michael Steele refers to the simple ex-
pectation by conditioning that we use to analyze the ruin probabilities
and expected duration. It is a more specific description for coin-tossing
games of the more general technique of expectation by conditioning.
Mathematical Ideas
Understanding a Stochastic Process
We start with a sequence of Bernoulli random variables,Y 1 ,Y 2 ,Y 3 ,...where
Yi= +1 with probabilitypandYi=−1 with probabilityq. We start with
an initial valueT 0 and setY 0 =T 0 for convenience. We define the sequence of
sumsTn=
∑n
i=0Yi. We are interested in the stochastic processT^1 ,T^2 ,T^3 ,....
It turns out this is a complicated sequence to understand in full, so we single
out particular simpler features to understand first. For example, we can look
at how many trials the process will experience until it achieves the value 0
ora. In symbols, considerN= min{n:Tn= 0,orTn=a}It is possible to
consider the probability distribution of this newly defined random variable.
Even this turns out to be fairly complicated, so we look at the expected value
of the number of trials,D=E[N]. This is a special case of a larger class of
probability problems calledfirst-passage distributionsforfirst-passage times.
Expected length of the game
Note that in the following we implicitly assume that the expected duration
of the game is finite. This fact is true, see below for a proof.
Theorem 2.The expected duration of the game in the classical ruin problem