Binomial Distribution - exact probability:

For the binomial distribution having n = 10, y = 0.4, find p (X ≤ 4) using equations and verify that equation gives a better approximation to the exact probability 0.6331.

Solution:

We have n = 10, p = 0.4, q = 0.6, np - 4, np q = 2,4 and √npq= 1.549. Hence equations give respectively

Φ ( 0/1.549 ) = Φ ( 0 ) = 0.5 and Φ ( 0.5/1.549 ) = Φ( 0.32 ) = 0.6255.

Thus above equation gives a better approximation.

In common, the normal approximation improves for a provide x as n increases and p approaches 0.5. For given n and p the approximation is better for x in the neighbourhood of n p than for x away from n p. Therefore, there is no rule of thumb that assures the quality of an approximation, even by the approximation is commonly found to be good while both np and nq are greater than 5.