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Chapter 1

Q. 1.1

Let t denote time in seconds and the random variable X(t), for any fixed t value, be the magnitude of the wind speed at a specified location at that time. Furthermore, let the family of X(t) random variables for all nonnegative t values be a stochastic process, {X(t)}, and let A be the event X(10)\leq 5m/s. . Review the amount of information needed to give complete probabilistic descriptions of the event A, the random variable X(t), and the stochastic process {X(t)}.


Verified Solution

All the probabilistic information about the event A is given by one number~its probability of occurrence. Thus, we might say that p=P(A) is that probability of occurrence, and the only other probabilistic statement that can be made about A is the almost trivial affirmation that 1-p=P(A^{C}) , in which A^{C}  denotes the event of A not occurring, and is read as “A complement” or “not A.”

We expect there to be many possible values for X(10). Thus, it takes much more information to give its probabilistic description than it did to describe A. In fact, one of the simpler comprehensive ways of describing the random variable X(10) is to give the probability of infinitely many events like A. That is, if we know P[X(10)\leq u]  for all possible u values, then we have a complete probabilistic description of the random variable X(10). Thus, in going from an event to a random variable we have moved from needing one number to needing many (often infinitely many) numbers to describe the probabilities.

The stochastic process {X(t): t \leq 0} is a family of random variables, of which X(10) is one particular member. Clearly, it takes infinitely more information to give the complete probability description for this stochastic process than it does to describe any one member of the family. In particular, we would need to know the probability of events such as [X(t_{1}) \leq u_{1}, X(t_{2}) \leq u_{2},…, X(t_{j}) \leq u_{j}] for all possible choices of j, t_{1},…,t_{j}, and u_{1},…, u_{j}.

If one chooses to extend this hierarchy further, then a next step could be a stochastic field giving the wind speed at many different locations, with the speed at any particular location being a stochastic process like {X(t)}.