Statistics - Poisson Distribution

Poisson conveyance is discrete likelihood dispersion and it is broadly use in measurable work. This conveyance was produced by a French Mathematician Dr. Simon Denis Poisson in 1837 and the dissemination is named after him. The Poisson circulation is utilized as a part of those circumstances where the happening's likelihood of an occasion is little, i.e., the occasion once in a while happens. For instance, the likelihood of faulty things in an assembling organization is little, the likelihood of happening tremor in a year is little, the mischance's likelihood on a street is little, and so forth. All these are cases of such occasions where the likelihood of event is little.

Poisson distribution is defined and given by the following probability function:

Formula

${P(X-x)} = {e^{-m}}.\frac{m^x}{x!}$

Where −

${m}$ = Probability of success.
${P(X-x)}$ = Probability of x successes.

Example

Problem Statement:

A producer of pins realized that on a normal 5% of his item is faulty. He offers pins in a parcel of 100 and insurances that not more than 4 pins will be flawed. What is the likelihood that a bundle will meet the ensured quality? [Given: ${e^{-m}} = 0.0067$]

Solution:

Let p = probability of a defective pin = 5% = $\frac{5}{100}$. We are given:

${n} = 100, {p} = \frac{5}{100} , \\[7pt] \ \Rightarrow {np} = 100 \times \frac{5}{100} = {5}$

The Poisson distribution is given as:

${P(X-x)} = {e^{-m}}.\frac{m^x}{x!}$

Required probability = P [packet will meet the guarantee]

= P [packet contains up to 4 defectives]

= P (0) +P (1) +P (2) +P (3) +P (4)

$ = {e^{-5}}.\frac{5^0}{0!} + {e^{-5}}.\frac{5^1}{1!} + {e^{-5}}.\frac{5^2}{2!} + {e^{-5}}.\frac{5^3}{3!} +{e^{-5}}.\frac{5^4}{4!}, \\[7pt] \ = {e^{-5}}[1+\frac{5}{1}+\frac{25}{2}+\frac{125}{6}+\frac{625}{24}] , \\[7pt] \ = 0.0067 \times 65.374 = 0.438$

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