5.1b Mean, Variance and Standard Deviation of a Binomial Distribution

5.1b Mean, Variance and Standard Deviation of a Binomial Distribution

2. Determine the mean, variance and standard deviation of binomial distribution.

If X is a binomial discrete random variable such that X ~ B (n, p),

then

Mean of X,

μ = n p

Variance of X,

σ^{2} = n p q

Standard deviation of X,

σ = \sqrt{n p q}

Example 1:

A Football club organises a practice session for trainees on scoring goals from penalty kicks. Each trainee takes 8 penalty kicks. The probability that a trainee scores a goal from a penalty kick is p. After the session, it is found that the mean number of goals for a trainee is 7.2.

Find the value of p.

Solution:

Mean = np

np = 7.2

8p = 7.2

p = 0.9

Example 2:

X is a binomial random variable such that X ~ B (n, p). If its mean and standard deviation are 90 and

3 \sqrt{7}

respectively, find the value of p and of n.

Solution:

\begin{array}{l} Mean = 90 \\ n p = 90 \\ Standard deviation = 3 \sqrt{7} \\ \sqrt{n p q} = 3 \sqrt{7} \\ n p q = 9 (7) \leftarrow Square both sides \\ n p q = 63 \\ (n p) q = 63 \\ 90 q = 63 \\ q = \frac{7}{10} \\ ∴ p = 1 - \frac{7}{10} = \frac{3}{10} \\ From n p = 90, \\ n (\frac{3}{10}) = 90 \\ n = 300 \end{array}

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