Poisson distribution is a discrete distribution i.e., X can take values 0, 1, 2,â€¦

**POISSION DISTRIBUTION**

In
a Binomial distribution with parameter *n* and *p* if the exact value of *n*
is not definitely known and if *p* is very small then it is not
possible to the find the binomial probabilities .Even if *n*
is known and it is very large, calculations are tedious. In such situations a
distribution called Poisson distribution is very much useful.

In
1837 French mathematician Simeon Dennis Poisson derived the distribution as a
limiting case of Binomial distribution. It is called after his name as Poisson
distribution.

i.
The number of trails â€˜*nâ€™* is indefinitely large i.e., nâ†’
âˆž

ii.
The probability of a success â€˜*p*â€™ for each trial is very small
i.e.,pâ†’ 0

*iii. np*= *Î»** *is finite

iv.
Events are Independent

A random variable X is said to follow a
Poisson distribution if it assumes only non-negative integral values and its
probability mass function is given by

(i)
Poisson distribution is a discrete distribution i.e., X can take values 0, 1,
2,â€¦

(ii)
p is small, q is large and n is indefinitely large i.e., p â†’0 q â†’1 and nâ†’3 and
np is finite

(iii)
Values of constants : (a) Mean = Î» = variance (b) Standard deviation = âˆšÎ» (c)
Skewness = 1/ âˆšÎ» (iv) Kurtosis =1/ Î»

(iv)
It may have one or two modes

(v)
If X and Y are two independent Poisson variates, X+Y is also a Poisson variate.

(vi)
If X and Y are two independent Poisson variates, X-Y need not be a Poisson variate.

(vii)
Poisson distribution is positively skewed.

(viii) It is leptokurtic.

i.
The
event of a student getting first mark in all subjects and at all the
examinations

ii.
The
event of finding a defective item from the production of a reputed company

iii.
The
number of blinds born in a particular year

iv.
The
number of mistakes committed in a typed page

v.
The
number of traffic accidents per day at a busy junction.

vi.
The
number of death claims received per day by an insurance company.

If
2% of electric bulbs manufactured by a certain company are defective find the
probability that in a sample of 200 bulbs (i) less than 2 bulbs are defective
(ii) more than 3 bulbs are defective. [*e*-4 = 0.0183]

Let
*X* denote the number of defective bulbs

In
a Poisson distribution 3*P*(*X*= 2) = *P*(*X* = 4). Find its parameter '*Î»*

**Example 10.13**

Find
the skewness and kurtosis of a Poisson variate with parameter 4.

*Solution:*

**Example 10.14**

If
there are 400 errors in a book of 1000 pages, find the probability that a
randomly chosen page from the book has exactly 3 errors.

*Solution:*

Let
*X* denote the number of errors in pages

**Example 10.15**

If
*X* is a Poisson variate with *P*(*X*=0) = 0.2725, find *P*(*X*=1)

*Solution:*

**Example 10.16**

The
probability of safety pin manufactured by a firm to be defective is 0.04. (i)
Find the probability that a box containing 100 such pins has one defective pin
. (ii) Among 200 such boxes, how many boxes will have no defective pin

*Solution:*

Let
*X* denote the number boxes with defective pins

Tags : Definition, Formula, Conditions, Characteristics, Solved Example Problems , 11th Statistics : Chapter 10 : Probability Distributions

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11th Statistics : Chapter 10 : Probability Distributions : Poission Distribution | Definition, Formula, Conditions, Characteristics, Solved Example Problems

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