When an independent trial is performed n times and “success” occurs with probability p on each trial, the distribution followed by the number of successes X is called the binomial distribution B(n, p).
P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ
Expected value and variance:
E(X) = np、V(X) = np(1−p)
Example ① Toss a fair coin 10 times. Find the expected value and variance of the number of heads X.
Solution:n=10, p=1/2.
E(X) = 5、V(X) = 10·(1/2)·(1/2) = 2.5
Solution:n=10, p=1/2.
E(X) = 5、V(X) = 10·(1/2)·(1/2) = 2.5
Key Point
- “Independent repeated trials” are the essence of the binomial distribution
Practice Problems
- Find the expected value and variance of B(20, 0.3).
Solutions and Explanations
- Solution:E=6、V=4.2