Approximation of Binomial distribution by Normal distribution:

Binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials.

Normal distribution is a continuous probability distribution that describes the distribution of a continuous variable.

When neither p nor q is very small but n is sufficiently large, the Binomial distribution can be closely approximated by the Normal distribution. This is known as the Normal approximation to the Binomial distribution.

The following are the reasons why Normal distribution can approximate Binomial distribution:

1. Central Limit Theorem: The Normal approximation to the Binomial distribution is based on the Central Limit Theorem. The Central Limit Theorem states that the sum of a large number of independent, identically distributed random variables tends to follow a Normal distribution.

2. Large Sample Size: When the sample size n is large, the Binomial distribution becomes more symmetric and bell-shaped, which makes it more similar to the Normal distribution.

3. Continuity Correction: Since the Binomial distribution is a discrete probability distribution and the Normal distribution is a continuous probability distribution, we need to apply a continuity correction to make the approximation more accurate.

The formula for the continuity correction is:

P(X = k) ≈ P(k - 0.5 < z="" />< k="" +="" />

where X is the Binomial random variable, k is the number of successes, Z is the standard Normal random variable, and P(k - 0.5 < z="" />< k="" +="" 0.5)="" is="" the="" probability="" of="" z="" falling="" between="" k="" -="" 0.5="" and="" k="" +="" />

In conclusion, when neither p nor q is very small but n is sufficiently large, the Binomial distribution can be closely approximated by the Normal distribution. This approximation is based on the Central Limit Theorem, the large sample size, and the continuity correction.