Theoretical Discrete Probability Distributions: Bernoulli, Binomial, Poisson, Geometric and Hypergeometric

For a binomial distribution $B(n, p)$:
$\text{Mean} = np = 8$
$\text{Variance} = npq = 6$

Dividing variance by mean:

The mode of a binomial distribution is determined by the value of $(n + 1)p$:

Since $(n + 1)p = 8.25$ is not an integer, the distribution is unimodal and the mode is the integral part $[(n + 1)p] = [8.25] = 8$.

For a Poisson distribution with parameter $\lambda$:
$\text{Mean} (\mu) = \lambda = 36$
$\text{Variance} (\sigma^2) = \lambda = 36 \implies \text{Standard Deviation } (\sigma) = \sqrt{36} = 6$

The Coefficient of Variation (C.V.) is defined as:

This represents a binomial experiment where:
Number of trials $n = 100$
Success = rolling an even number $\{2, 4, 6\}$ on a 6-sided die $\implies p = \frac{3}{6} = \frac{1}{2} = 0.5$
Failure probability $q = 1 - p = 0.5$

The variance of a binomial distribution is:

This follows a Binomial distribution $B(n = 10, p = 0.16)$:
Probability of failure $q = 1 - p = 1 - 0.16 = 0.84$
$\text{Mean} = np = 10 \times 0.16 = 1.6$
$\text{Variance} = npq = 10 \times 0.16 \times 0.84 = 1.344$
$\text{Standard Deviation} (\sigma) = \sqrt{\text{Variance}} = \sqrt{1.344} \approx 1.1593$