Binomial Distribution

Binomial Distribution

How to find the expected value, variance, and the standard deviation from the binomial distribution data: formulas, examples, and their solutions.

Formula

For a binomial experiment with a very large n, E(X) = np, V(X) = npq, sigma(X) = sqrt(npq). n: Number of trials, p: Probability of 'want', q = 1 - p.

For a binomial experiment
with a very large n:

E(X) = np

Expected value

V(X) = npq

σ(X) = √V(X)
= √npq

Variance, Standard deviation

Example 1

A coin is tossed 100 times. 1. Find the expected value of the number of 'head's. 2. Find the variance of the number of 'head's. 3. Find the standard deviation of the number of 'head's.

To use the formulas,
write n, p, and q.

n = 100 (100 trials)
p = 1/2 (Probability of 'head' shown)
q = 1 - 1/2 = 1/2 (Probability of 'head' not shown)

Probability of (not A, Complement)

E(X) = 100⋅(1/2)

V(X) = 100⋅(1/2)⋅(1/2)

You can find σ(X)
by directly square rooting the variance (√25)
or by using the formula (√100⋅(1/2)⋅(1/2)).

Example 2

A fair die is tossed 360 times. 1. Find the expected value of the number of '1'. 2. Find the variance of the number of '1'. 3. Find the standard deviation of the number of '1'.

To use the formulas,
write n, p, and q.

n = 360 (360 trials)
p = 1/6 (Probability of '1' shown)
q = 1 - 1/6 = 5/6 (Probability of '1' not shown)

Probability of (not A, Complement)

E(X) = 360⋅(1/6)

V(X) = 360⋅(1/6)⋅(5/6)

You can find σ(X)
by directly square rooting the variance (√50)
or by using the formula (√360⋅(1/6)⋅(5/6)).