# Vsub Segment 41

## Calculating

### Poisson and Exponential

1. Show that the waiting times (times between events) in a Poisson process are Exponentially distributed. (I think we've done this before.)

This follows from the definition of a Poisson process. A Poisson process (is a stochastic process) that counts the number of events and the time that these events occur in a given interval. The time between each pair of consecutive events has an exponential distribution. Mathematically:

The number of events (of a constant rate Failed to parse (unknown error): \lambda ) that occur in a given time interval is a Poisson random variable given by

   $e^{-\lambda \tau} (\lambda \tau)^k}{k!$



N(t) denotes the number of events that have occured in time t.

The waiting time between two events: we want the distribution of Failed to parse (unknown error): \tau (when k=1 in the above definition). Let the arrival times of events in the Poisson process be given by Failed to parse (unknown error): T_1, T_2, \ldots . Then, we can say that the probability of the $th$ event occurring after time t is equal to the probability of there having occurred fewer than Failed to parse (unknown error): n events in the time upto t. Mathematically,

 Failed to parse (unknown error):  P(T_n > t) = P (N(t) < n)



Now, consider the first event (n=1):

 $e^{-\lambda t} (\lambda t)^{0}}{0!} = e^{-\lambda t$



This shows that the waiting time (t) until the first arrival has an exponential distribution. Similarly,

 $n-1}) > t) = P[(N(t_n) - N(t_{n-1})) = 0] = \frac{e^{-\lambda t} (\lambda t)^{0}}{0!} = e^{-\lambda t$



Hence, the waiting times between events in a Poisson process are exponentially distributed (and the expected wait time is Failed to parse (unknown error): 1/\lambda )

### Wait times

2. Plot the pdf's of the waiting times between (a) every other Poisson event, and (b) every Poisson event at half the rate.

### Poisson and Gamma

3. Show, using characteristic functions, that the waiting times between every Nth event in a Poisson process is Gamma distributed. (I think we've also done one before, but it is newly relevant in this segment.)