原创 Probability of a random jitter event

RJ is typically quantified as an rms value equal to σ of the Gaussian formula. Since Gaussian distributions are unbounded, a peak-to-peak value of RJ requires a measure of probability of bit errors, or BER (bit error ratio). For a typical specification of BER = 10^-12, (1 error for every 10¹² bits) the peak-to-peak value of RJ is closely approximated by 14σ, where σ is one standard deviation.

Figure 1 shows a plot of the familiar Gaussian distribution. We are often concerned with probability in any given area of this bell curve. Table 1 shows the probability of events occurring in the peak and tail regions for different spans of σ.

Figure 1. Gaussian distribution.

Table 1. Probability of peak and tail areas vs. number of σ.

 

At ±6σ, we see that the probability of the tails is about two in a billion. For ±7σ, the probability of the tails is about 2.6 in a trillion, which is roughly how we came to use 14σ to bound RJ at BER = 10^-12. Even though we frequently talk about probabilities of less than one in a trillion, few people actually have an intuitive grasp of such large/small numbers. I came across a very interesting example to illustrate Gaussian distributions using the distribution of height in American men.

The US has approximately 300 million people, out of whom approximately 100 million are non-acromegalic adult men (i.e., no abnormal pituitary problems affecting their height). The mean height of men in the US is about 5'9" with a standard deviation of 3". Knowing how Gaussian distributions work, we can estimate how many men are in each height bracket, shown below in Table 2.

Table 2. The probability of the height of men.

 

We see that about 68 million men are within ±1σ of the mean, which is between 5'6" and 6' tall. We refer to this range as average height. We think of men between 6' and 6'3" as being relatively tall, but there is still a large population of them, about 13.6 million or 13.6 percent. The numbers drop off very quickly for increasing heights, as expected.

What is interesting is that Gaussian statistics tell us that there should be no men in the US who are above 7'3" tall. But, many sports fans can immediately point out that Yao Ming is 7'6". Someone of Yao Ming's height is statistically unlikely in the normal height distribution of American men.

So, why is Yao Ming Chinese? Because the US does not have enough adult men to realize someone who is greater than 6σ in height! Instead, Yao Ming was born in China, where the current population of adult men is about 500 million, which is five times more likely to grow a 6σ person. Furthermore, the mean height of men in China is approximately 5'7", which makes Yao Ming's stature even more statistically rare!

Yao Ming, basketball player for the Houston Rockets (retired).

 

Many readers will quickly point out that India also has a population comparable to China's. Is there at least one Indian man who is 6σ above the mean in height? The answer is a resounding yes! Sunil Chaudhary currently claims to be the tallest living man in India at a height of 7’6", a number that neatly fits into our understanding of 6σ in Gaussian distributions.

So, in jitter analysis, when we are looking for that less-than-one-in-a-trillion event that determines our BER compliance, or when we test for violations with a PRBS31 (2³¹−1) pattern -- 2.4 billion bits long -- I like to tell people I'm searching for the "Yao Ming of jitter."

Daniel Chow