Tektronix 071-0590-00 user manual Poisson Error Process, Pn = rT n e−rT . n

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Poisson Error Process

A Poisson process is one in which events are not dependent on each other, and conditions causing the events don't change with time. Raindrops hitting a skylight is an example of a Poisson process. The impact of one raindrop doesn't affect the arrival time of another. If we know the average rate r of a Poisson error process, then we have completely characterized it. In particular, if errors are measured for a period T, the probability of measuring n errors is given by

p(n) = ( rT) n e−rT . n!

(For a proof of this, see Probability, Random Variables, and Stochastic Processes by Papoulis, McGraw- Hill, 1965.)

For example, for r = 5 / hr and T = 1 hr, the probability of measuring n errors in one hour is plotted in Figure S-1a. It is most probable that n = 4 or 5 errors will be measured. The probabilities for all the n sum to one.

If the one-hour test is repeated many times, the mean (or expected) number of errors is

µ= rT.

For the case r = 5 / hr and T = 1 hr, on average µ = 5 errors will be measured. As the test is repeated many times for the same r and T, the standard deviation of the measurement n is given by

σ =		μ =			rT . (see Papoulis for a proof of this).
In this case σ =			5 = 2.24, which is 45% of μ (see Figure S-1b).
0.18												0.06
0.16												0.05
0.14
0.12									σ = 0.45μ			0.04
0.1												p( n )										σ = 0.14μ
p (n )												0.03
0.08							σ
			σ
0.06
												0.02
0.04																		σ σ
0.02												0.01
					μ
0 0																		μ
	1	2	3	4	5	6	7	8	9	10	11	12 13	0	10	20	30	40	50	60	70	80	90	100	110	120 130
							n						0
(S-1a)												(S-1b)								n

Figure S-1a. Probability p(n) of n errors in one hour for an average of 5 per hour. The error process is Poisson. About 68% of the time n is no more than σ away from the mean value μ = 5.

Figure S-1b. Probability p(n) of n errors in ten hours for an average of 5 per hour. Because the mean number of errors is larger (μ = 50), σ is now smaller in relation to μ.

If the measurement time is increased to T = 10 hr, then µ = 5 × 10 = 50 errors, and the curve of p(n) becomes much "tighter", as shown in Figure B. Now σ = 50 = 7.07, which is only 14% of μ. About 68% of the area under the curve lies between n = μ − σ and μ + σ, indicating that 68% of the time n will lie in this range. If we consider 68% to be "most of the time," then we can write the "bounds" on n as

µ− σ < n < µ + σ.

For r = 5 / hr and T = 10 hr, the range is ±14%, which is better than the range of ±45% for T = 1 hr. However, the tighter range came at the cost of ten times the test time.

The estimated error rate r' = n / T is correspondingly "bounded" by

B-42 GB1400 User Manual