How Do You Approximate a Binomial?

Trying to understand how how Do You Approximate a Binomial matters today? Find the full explanation and key takeaways.

The binomial approximation is useful for approximately calculating powers of sums of a small number x and 1. ( 1 + x ) α ≈ 1 + α x .

Is the size of N large enough to use the normal approximation?

Verify whether n is large enough to use the normal approximation by checking the two appropriate conditions. For the above coin-flipping question, the conditions are met because n ∗ p = 100 ∗ 0.50 = 50, and n ∗ (1 – p) = 100 ∗ (1 – 0.50) = 50, both of which are at least 10. So go ahead with the normal approximation.

Which is a good estimate of the binomial distribution?

In the above graphic, the binomial distribution shown resulted from n = 20 trials with probability of success p = 0.50. In this case, n p = n q = 10 ≥ 5, and we can see the approximation is a good one.

Is the binomial approximation related to Bernoulli’s inequality?

The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli’s inequality, the left-hand side of the approximation is greater than or equal to the right-hand side whenever. x > − 1 {\\displaystyle x>-1}. and. α ≥ 1 {\\displaystyle \\alpha \\geq 1}. .