If n is large, the evaluation of the binomial distribution can involve considerable computation. In such a case a simple approximation to the binomial probability could be considerable use. The approximation of binomial when n is large and p is close to zero is called the Poisson Distribution Mean.
Definition:
A random variable is said to follows Poisson distribution if it assumes only non-negative values and its probability mass function is given by
P(x,lambda ) =P(X=x) = elambda lambda x / X!
x=0, 1, 2…
O
otherwise
lambda is known as parameter of Poisson distribution .X~p() denotes it.
Characteristic function of Poisson distribution
phi x ( t ) =E[ei t x ]
=sum_(x=0)^oo ei t x e-lambda lambda x / x!
= e-lambda sum_(x=0)^oo ei t x lambda x / x!
= e-lambda [1+(lambda eit ) + (lambda eit)2 / 2! +(lambda ei t)3 /3! + ............. ]
=e-lambda elambda eit
= elambda (ei t -1)
Additive property of Poisson distribution:
Independent Poisson variate is also a Poisson variate xi (i=1,2,......n). xi follows Poisson with parameter lambda i.
xi ~ P(lambda i ) (i=1,2.....n)
then sum_(i=1)^n xi ~ P(sum_(i=1)^n lambda i )
Proof:
Mxi(t) = elambda i(e^t -1)
Mx1+x2+x3+.....Xn(t) = Mx1 (t) Mx2 (t). . . Mxn (t)
= elambda 1(e^t-1). elambda 2(e^t-1) ..........elambda n(e^t-1)
=e(lambda 1 +lambda 2 +lambda 3+.......lambda n )(et-1)
=esum_(i=1)^n lambda i(et-1)
M.G.F of Poisson distribution:
Mx ( t ) = E [ etx ]
= sum_(x=0)^oo etx e-lambda lambda x / x!
= e-lambda sum_(x=0)^oo etxlambda x / x!
= e-lambda [ 1+ et lambda + (lambda et )2 /2! + ( (lambda et )3 / 3! +...... ]
=e-lambda elambda
= elambda (et -1)
Applications of Poisson distribution
The following are some instance where the distribution is applicable
Number of deaths from a disease
Number of suicide reported in a particular city.
Number of defective materials in packing manufactured by good concern.
Number of printing mistakes at each page of the book.
Number of air accidents in some unit of the time.
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