Question

Suppose X follows the Binomial Distribution with n=1000 and p=0.002. Use the Poisson approximation to determine the probability that X is at least 2.

Answer 1

X follows binomial distribution with n=1000 and p=0.002.

Here,

np=1000*0.002=2

which is a very small but finite value.

So,we can use the poisson approximation to determine probability here.

As np=2,we can safely say that X approximately follows poisson distribution with parameter 2.

So,pdf of X is

f_X(x)=e^-2*(2^x)/fact(x) , when x=0,1,2,....

P(X is at least 2 )

=P(X>=2)

=1-P(X<2)

=1-P(X=0)-P(X=1)

=1-e^-2*(2⁰)/fact(0)-e^-2*(2¹)/fact(1)

=1-e^-2-2e^-2

=1-3e^-2

^=1-0.406

^=0.594

^{So,required probability is 0.594.}