For the discrete distribution, the Pearson's coefficient of skewness β 2is always:
β2 > 1
For any distribution with finite non-zero variance, the Cauchy–Schwarz inequality applied to \((X-\mu)^2\) gives \(\mu_4 \ge \mu_2^2\), hence \(\beta_2 = \dfrac{\mu_4}{\mu_2^2} \ge 1\).
For a discrete distribution (taking values at distinct mass points), \((X-\mu)^2\) is not constant, so the inequality is strict:
\[\beta_2 = \dfrac{\mu_4}{\mu_2^2} > 1\]
Hence for a discrete distribution, \(\beta_2\) is always greater than 1.
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