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Poisson formula and an example of solving a problem

Lecture



Bernoulli's formula is convenient for calculations with only a relatively small number of tests. Poisson formula and an example of solving a problem . For large values Poisson formula and an example of solving a problem use the formula uncomfortable. Most often, in these cases, the Poisson formula is used. By the way, this formula is determined by the Poisson theorem.

Note that the theorem. If the probability Poisson formula and an example of solving a problem event occurrence Poisson formula and an example of solving a problem in each test is constant and small, and the number of independent tests Poisson formula and an example of solving a problem large enough, the probability of an event Poisson formula and an example of solving a problem smooth Poisson formula and an example of solving a problem times approximately equal

Poisson formula and an example of solving a problem , (3.4)

Where Poisson formula and an example of solving a problem .

Evidence. Let given the probability of occurrence Poisson formula and an example of solving a problem in one test Poisson formula and an example of solving a problem and the number of independent trials Poisson formula and an example of solving a problem . Denote Poisson formula and an example of solving a problem . Where p = λ / n Poisson formula and an example of solving a problem . Substitute ϶ᴛᴏ expression in the Bernoulli formula:

Poisson formula and an example of solving a problem

For a sufficiently large !! n ,, and relatively small !! m ,, all brackets, with the exception of the penultimate one, can be taken equal to one, i.e.

Poisson formula and an example of solving a problem

Given that n is sufficiently large, the right side of this expression can be considered when n-> oo, i.e. find limit

Poisson formula and an example of solving a problem

Then we get

Poisson formula and an example of solving a problem (3.5)

Example. The company manufactured and sent to the customer 100,000 bottles of beer. The likelihood that a bottle might be a bat is 0.0001. Find the probability that there will be exactly three and exactly five broken bottles in the shipment.

Decision. Given: n = 100000, p = 0.0001, m = 3 (m = 5).

Find Poisson formula and an example of solving a problem = 10

We use the Poisson formula

Poisson formula and an example of solving a problem


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis