- Example: use following steps to generate uniformly distributed
random numbers between 1/4 and 1.
**Step 1.**- Generate a random number
*R* **Step 2a.**- If , accept
*X = R*, goto Step 3 **Step 2b.**- If , reject
*R*, return to Step 1 **Step 3.**- If another uniform random variate on [1/4, 1] is needed, repeat the procedure begining at Step 1. Otherwise stop.

- Do we know if the random variate generated using above methods is
indeed uniformly distributed over [1/4, 1]? The answer is Yes. To
prove this, use the definition. Take any
,

which is the correct probability for a uniform distribution on [1/4,1]. - The efficiency: use this method in this particular example,
the rejection probability is 1/4 on the average for each number
generated. The number of rejections is a geometrically
distributed random variable with probability of ``success''
being
*p = 3/4*, mean number of rejections is*(1/p - 1) = 4/3 - 1 = 1/3*(i.e. 1/3 waste). - For this reason, the inverse transform (X = 1/4 + (3/4) R) is more efficient method.

**Poisson Distribution**- pmf

where*N*can be interpreted as the number of arrivals in one unit time. - From the original Poisson process definition, we know the
interarrival time are exponentially distributed with
a mean of , i.e. arrivals in one unit time.
- Relation between the two distribution:

if and only if

essentially this means if there are*n*arrivals in one unit time, the sum of interarrival time of the past*n*observations has to be less than or equal to one, but if one more interarrival time is added, it is greater then one (unit time). - The s in the relation can be generated from uniformly
distributed random number
, thus

both sides are multiplied by

that is

- Now we can use the Acceptance-Reject method to generate Poisson
distribution.
**Step 1.**- Set
*n = 0, P = 1*. **Step 2.**- Generate a random number and replace
*P*by . **Step 3.**- If
, then accept
*N = n*, meaning at this time unit, there are*n*arrivals. Otherwise, reject the current*n*, increase*n*by one, return to Step 2.

- Efficiency: How many random numbers will be required, on the average,
to generate one Poisson variate,
*N*? If*N = n*, then*n+1*random numbers are required (because of the (n+1) random numbers product).

- Example 9.10 on page 346, Example 9.11 on page 347
- When is large, say , the acceptance-rejection
technique described here becomes too expensive. Use normal distribution
to approximate Poisson distribution. When is large

is approximately normally distributed with mean 0 and variance 1, thus

can be used to generate Poisson random variate.

- pmf