Abstract:
There are numerous applications of exponential continuous random variables and stochastic processes. However, the theoretical analysis and application of these processes, in particular when the random variable is both discrete and truncated, is not completed. This Report presents firstly a review of exponential continuous density functions, both un-truncated and truncated. Then the discrete density function is derived and expressed in terms of Dirac’s delta functions. For this case, the mean and variance are derived and analyzed. The necessity of having truncated discrete density function, from the application point of view in communication systems, is explained and related density and distribution functions are derived. For these functions, the mean and variance expressions are expressed as functions of the length of the defined truncation interval and compared with related moments of the continuous truncated density function. The important advancement is achieved by deriving the truncated discrete density functions and expressing them in terms of Dirac’s delta and unit step functions. How to design a discrete truncated exponential stochastic process is illustrated on several examples.