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app5_rain_norain - Application Example 5(Bernoulli Trial...

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Application Example 5 (Bernoulli Trial Sequence and Dependence in Binary Time Series) IS THE SERIES OF RAINY/NON-RAINY 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 so-called 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/non-rainy 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 |i-j|.
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The joint distribution of the daily rain/no-rain 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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