Waiting time distribution

Now let us consider the distribution function of , which is the waiting time for the occurrence of  number of events (which is deterministic) in a probabilistic time frame, say .

Now the  events occurs prior or at time  iff the number of events occurring by time  is at least , i.e., the following holds true, i.e.,

Thus

i.e., , where and

Thus:

Thus:
                                                   

which implies

Using the fact that ,  are i.i.d exponential random variables with mean  we show  has gamma distribution with  and  as the parameters, hence the probability density function is of the form .

Note

• Remember that the exponential distributed discussed here is also referred to as the gamma distribution wit h parameters  and 1.
  
  
• From the reproduction property of gamma distribution one notes that if , then .