When looking at the Poisson distribution we said that if we

When looking at the Poisson distribution, we said that if we have a binomial experiment with a large n and a small p, the binomial pmf can be approximated by a Poisson pmf with ? Mu =np. Let??s verify this proposition: Suppose that 2% of all copies of a particular book are bound incorrectly. Let X be the number of copies that are not bound correctly in a sample of 100. Use both the binomial and Poisson distributions to calculate P(X

Solution

Binomial distribution with n=100 and p=0.02

P(X=x)=100Cx*(0.02^x)*(0.98^(100-x)) for x=0,1,2,...,100

So P(X<3) = P(X=0)+P(X=1)+P(X=2)

=100C0*(0.02^0)*(0.98^(100-0))+...+100C2*(0.02^2)*(0.98^(100-2))

=0.6766856

-----------------------------------------------------------------------------------------------------------------------------

Poisson distribution with mean=n*p=100*0.02 =2

P(X=x)=(2^x)*exp(-2)/x!

So P(X<=3)= P(X=0)+P(X=1)+P(X=2)

=(2^0)*exp(-2)/1+....+(2^2)*exp(-2)/2!

=0.6766764

-----------------------------------------------------------------------------------------------------------------------------

Yes, because both probability are closed.