Application Example 5
(Bernoulli Trial Sequence and Dependence in Binary Time Series)
IS THE SERIES OF RAINY/NONRAINY DAYS A BERNOULLI
TRIAL SEQUENCE?
Note: The shaded text in this note involves the concept of correlation function for a random
sequence. This concept will be encountered later in the course. In a fist reading, you may skip
that text.
The concept of dependent random variables finds an important application in socalled time
series, which are models of how a random quantity varies in time.
In the case when time is
discrete or discretized, as for example happens when one considers variables with daily, monthly
or annual values (daily close of the stock market, monthly average temperatures, annual sales,
score of i
th
baseball game, etc.), the time series is simply a discrete sequence of random variables
X
i
.
An important issue in modeling such sequences is the probabilistic dependence among
different variables.
Here we illustrate the concepts of dependence in time series by considering the simplest case,
which is a series of indicator variables {I
i
, i = 0, ± 1, ±2, ...}. Such variables can be used to
indicate whether or not an event of interest occurs (I
i
= 1) or does not occur (I
i
= 0) at “time” i.
For example, we can take the event of interest to be the fact that day i is rainy. Again to keep the
illustration simple, we consider the case when the sequence of rainy/nonrainy days is stationary.
Stationarity means that the sequence has everywhere the same statistical properties. Hence, the
probability P[I
i
= 1] does not depend on i, the probability P[(I
i
= 1)
∩
(I
j
= 1)] depends only on the
separating distance i  j, and so on. The assumption of stationarity is realistic in many cases and
greatly simplifies the characterization of a time series.
i
, i = 0, ± 1, ±2, ...} is that the correlation
function
ρ
ij
An implication of stationarity for a random sequence {I
depends only on the time lag ij.
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The joint distribution of the daily rain/norain indicators I
i
and I
j
has 4 probability masses:
• a probability mass p
00
at (0,0), which gives the probability that both days are dry,
• a probability mass p
11
at (1,1), which gives the probability that both days are wet,
• probability masses p
01
and p
10
at (0,1) and (1,0), which give the probability that day i is dry
and day j is wet and, viceversa, that day i is wet and day j is dry.
These four probabilities must add to unity.
Moreover, due to stationarity, p
01
= p
10
, meaning that
the relative frequency of the two “transitions” (dry
→
wet and wet
→
dry) must be the same.
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 Fall '07
 DonHill
 Probability theory, ij, p11

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