 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

Expected value

V(X) = npq

σ(X) = √V(X)
= √npq

Variance, Standard deviation

Example 1 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 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)).