Unification of probability theory on time scales
© Ufuktepe; licensee Springer 2012
Received: 14 August 2012
Accepted: 27 November 2012
Published: 10 December 2012
The theory of time scales was introduced by Stefan Hilger in his PhD thesis in 1988 in order to unify continuous and discrete analysis. Probability is a discipline in which appears to be many applications of time scales. Time scales approach to probability theory unifies the standard discrete and continuous random variables. We give some basic random variables on the time scales. We define the distribution functions on time scales and show their properties.
MSC: 46N30, 60B05.
Time scale calculus has received a lot of attention [1, 5–7]. In recent years there have been many research activities about applications of time scales. Probability theory is an ideal discipline for applications of time scales since random variables and distributions functions can be described with either discrete or continuous cases.
We give a brief introduction to measure theory on time scales introduced by Guseinov  in Section 2. We give the discussion of our original probability results in Section 3. In Section 4 we study the discrete random variables, i.e., binomial, Poisson, geometric, and negative binomial random variables on a discrete time scale . In Section 4.5 we define uniform random variables on the time scale, and we give the definition of Gaussian bell in Section 5.
2 Measure on time scales
The Riemann Δ integral has been introduced by Guseinov in , the Δ measure and the Lebesgue Δ integral were introduced by Guseinov in  and studied by Cabada , Ufuktepe and Deniz in , and Rzezuchowski in . In this section we set out basic concepts of Δ and ∇ measures.
be the family of all left closed and right open intervals of . Then is a semiring. Here . is a set function which assigns to each interval its length: . So, if is a sequence of disjoint intervals in , then .
where . If there is no such covering of E, then .
Since we always have , we see that E is Δ-measurable iff for each A we have .
If E is Δ-measurable, then is also Δ-measurable. Clearly, Ø and are Δ-measurable.
Lemma 2.2 If and are Δ-measurable, so is .
Let be a family of Δ-measurable sets.
Corollary 2.3 is a σ algebra.
Definition 2.4 The restriction of to is called the Lebesgue Δ-measure and denoted by .
where is understood as an empty set, then such that is a countably additive measure. Then is the set of ∇-measurable sets and is the Lebesgue ∇-measure on .
Proof . □
Proposition 2.6 (Properties of )
(ii) If , then ;
Proof Similar to the proof in . □
Theorem 2.7 For each , the single point set is Δ-measurable and its Δ-measure is given by .
Proof Case 1. Let be right scattered. Then . So, is Δ-measurable and .
which is the desired result since is right dense. □
Every kind of interval can be obtained from an interval of the form by adding or subtracting the end points a and b. Then each interval of is Δ-measurable.
Theorem 2.8 If and , then
(iii) If , then and .
Proof . □
Theorem 2.9 For each , the ∇-measure of the single point set is given by .
Proof Similar to case. □
Theorem 2.10 If and , then
(iii) If , then and .
Proof The equalities can be obtained by the same technique with case. □
Lemma 2.11 , where is the outer measure of E.
(i) and , then .
(ii) Let and , then .
3 Probability on time scales
if , then .
Definition 3.1 Let be a time scale and be a field of subsets of . Suppose that is a Δ-measure defined on . Then is a probability measure if . In this case, the triple is called a Δ-probability space.
is called ∇-probability of A.
Proposition 3.3 and are probability functions.
The proof of is similar. □
which is equivalent to the counting probability.
Proposition 3.4 For any , we have if .
4 Discrete random variables on time scales
Definition 4.1 A random variable is a real-valued function defined on .
In this section we consider the binomial, Poisson, geometric, and negative binomial random variables on , where .
4.1 Binomial random variable on
where , is a grainness function, and is called a binomial random variable on the time scale. Since , we take h instead of .
Example Consider a jury trial in which it takes eight out of twelve juror groups to convict; that is, in order for the defendant to be convicted, at least eight of the juror groups must vote him guilty. Also, consider each group consists of three members. If at least two of three members vote that the defendant is guilty, then the decision of the group is guilty. If we assume that each juror group acts independently and each person makes the right decision with probability θ, what is the probability that the jury renders a correct decision?
If α represents the probability that the defendant is guilty, then is the desired result.
While evaluating the expected value and the variance of the discrete random variables, we will make use of the following proposition.
Proposition 4.2 If is a discrete random variable that takes on one of the values , , with the respective probabilities , then for any real-valued function g, .
Remark 4.3 When we take , then the time scale is a set of natural numbers and the expected value is as in the classical probability theory.
Remark 4.4 When we take , then the time scale is a set of natural numbers and the variance is as in the classical probability theory.
4.2 Poisson random variable on
then is a probability mass function.
Example An energy company produces batteries and sells five in a box. The probability that a battery is defective is 0.1. We assume if a box contains at least two defective batteries, then this box is also defective. Find the probability that a sample of ten boxes contains at most one defective.
4.3 Geometric random variable on
Equation (4.5) follows because in order for to equal , it is necessary and sufficient that the first trial groups are failures and the th trial group is a success. Equation (4.5) then follows, since the outcomes of the successive trial groups are assumed to be independent.
it follows that with probability one, a success group will eventually occur.
Definition 4.6 Any random variable whose probability mass function is given by Equation (4.5) is said to be a geometric random variable with the parameter .
4.4 Negative binomial random variable on
Example A student takes multiple choice exams which have five questions with three choices. The student is successful if he/she gives at least three correct answers in an exam. What is the probability of the third success of the student in the tenth exam by guessing?
where and .
4.5 Uniform random variable on the time scale
Remark 4.7 If we take the left closed and right open interval on our time scale T such that , then the integral over this set , and also, if we take right and left open intervals and since a is right dense, then our result is the same .
5 Gaussian bell on time scales
where must be −t in the case . Erbe and Peterson  defined the Gaussian bell on the time scale as follows.
which implies , .
Mathematical induction is used for showing Equation (5.6).
By using an exponential function on time scales, we can define an exponential probability density function in a general case and we can define a moment generating function by using Laplace transformations on time scales. Then future works can be stochastic processes on time scales and stochastic dynamic equations.
To my wife
I would like to thank TUBITAK and the referees for their support and their valuable comments.
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