POISSON DISTRIBUTION: LIMIT OF BINOMIAL DISTRIBUTION FOR RARE OCCURRENCE


Rare successes: Probability p extremely small but sample size n extremely large, such that binomial mean = np = a is finite. Since p is so small, we can expect that x success will also be small compared to sample size n. The Poisson distribution is also used to compute the probability of the number of "Poisson events" during a given time interval or within a specified region of space.

We already have defined the binomial distribution as:

For the Poisson distribution, however, we:

  1. Reduce the permutation factor, since x << n.

  2. Combine factors of like powers.

  3. Note that because p « 1 and x is not large, we can approximate (1 - p) x = 1.

  4. Having mean np = a we can rewrite n = a/p so

  5. Limit as p 0 yields exponential

  6. The result is the Poisson distribution (one parameter: a).

    1. In reliability, the parameter a = t.

    2. is the mean number of events per unit time.

COMPARISON OF BINOMIAL AND POISSON

The probability density function (pdf) for the binomial distribution is:

B (Xi = x;n, p) = [n!/X!(n - x)!] p x q (n-x)

  • Mean: ¼ = np = a

  • Variance: ƒ 2 = npq = aq

  • Two Parameters: n and p

    • or

    • a(= np) and q = (1- p)

On the other hand, the probability function for the Poison distribution is:

where a = n p and q = (1 - p) approx. 1.

  • Mean: ¼ = a

  • Variance: ƒ 2 = a

  • One parameter: a( = np)

Example 5
start example

A compressor manufacturer has a record of 50 defects per 1000 produced, which corresponds to a defect percentage of 5%. Since the probability of successfully observing a defect is small (p = 0.05), then we can assume a Poisson distribution.

Determine probability of observing defects in batch of n = 10:

Poisson parameter: a = np = 10(0.05) = 0.5

Probability of exactly x = 0 failures:

Probability of exactly x = 1 failure:

Probability of exactly x = 2 failures:

Probability of exactly x = 3 failures:

Figure 16.32 shows the Poisson distribution for each of the failures.

click to expand
Figure 16.32: Poisson distribution for the four failures.
click to expand

Special comments:

  1. The mean is equal to the parameter a; as a result, the smaller the "a's," the more skewed the probability density function (pdf) is to the left.

    click to expand
  2. The standard deviation is equal to the square root of "a"; hence, increasing the value of "a" also increases the standard deviation.

  3. The larger the value of "a," the more the Poisson distribution approaches the normal distribution.

end example
 



Six Sigma and Beyond. Statistics and Probability
Six Sigma and Beyond: Statistics and Probability, Volume III
ISBN: 1574443127
EAN: 2147483647
Year: 2003
Pages: 252

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