For example, in tossing a coin, if head occurs, occurrence of tail cannot take place. Conditional and Joint probability. Suppose that we contemplate two experiments A and B with outcomes A1, A2, ... and B1, B2, ... . The probability of outcome Bk, given that A, is known to have occurred is called conditional probability given as
Similarly, the probability of outcome A,, given that Bk is known to have occurred is given as
where P(Aj, Bk) is called joint probability, that is the joint occurrence of Aj and Bk. From equation 8.11 and 8.12, P(A,, Bk) is given as
P(A,, Bk) = P(Bk |A,) P(Aj) = P(A,, Bk) P(Bk) ...(8.13)
Sub 8.13 in equation 8.12 gives
This result is known as Bayes' theorem.
If the outcome of Bk does not depend at all on which outcome A. accompanies it, we say that the outcomes A. and Bk are independent. When outcomes are independent, the probability of a joint occurrence of particular outcomes is the product of the probabilities of the individual independent outcomes. It is given as
This result may be extended to any arbitrary number of outcomes. Thus
Random variables and Random process. Subscribers generates calls in random manner. The call generation by the subscribers and therefore the behaviour of the network or the switching system is described as a random process. It is also referred as stochastic process. In random process, one or more quantities vary with time in such a way that the instantaneous values of the quantities are not determinable but are predictable with certain probability. The quanties are called random variables.
In telecommunication system, telephone traffic is referred as random process and the number of simultaneous active subscribers and simultaneous busy servers are assumed as random variables. The variation of traffic over a period of time (30 minutes or 60 minutes) is a typical random process. The random process may be discrete or continuous. In telecommunication, the variable representing the number of simultaneous calls is discrete. Thus, in our modelling we use discrete state stochastic processes.
Definition of statistical terms :
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