Statistical Methods & Experimental Designs

By Karl Pearson's moment inequality derived from the Cauchy-Schwarz inequality for any probability distribution:

where $\beta_1 = \frac{\mu_3^2}{\mu_2^3}$ is the measure of skewness and $\beta_2 = \frac{\mu_4}{\mu_2^2}$ is the measure of kurtosis.
Since $\beta_1 \ge 0$ (being a ratio involving a squared real quantity), it strictly follows that:

Option analysis:
(A) For any discrete distribution, kurtosis satisfies $\beta_2 \ge 1$, where equality is attained by a symmetric two-point distribution (e.g. $\text{Bernoulli}(0.5)$).

When a single six-faced die is thrown, the number of possible outcomes is 6. When two independent unbiased dice are thrown simultaneously, the total number of outcomes in the sample space $S$ is given by the fundamental principle of counting:

The outcomes range from $(1,1), (1,2), \dots, (6,6)$.