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CSS Statistics QUESTION #1975
Question 1
The first three central moments of a distribution are \(\mu_1' = 1\), \(\mu_2' = 16\), and \(\mu_3' = -40\), all computed about the value \(a = 2\). The third central moment \(\mu_3\) about the mean equals:
  • \(-86\)✔️
  • \(-40\)
  • \(76\)
  • \(-96\)
Correct Answer Explanation
Using the relation \(\mu_3 = \mu_3' - 3\mu_2'\mu_1' + 2(\mu_1')^3\): Mean \(= a + \mu_1' = 2+1 = 3\), so \(d = \mu_1' = 1\). Then \(\mu_3 = -40 - 3(16)(1) + 2(1)^3 = -40 - 48 + 2 = -86\). The third central moment is \(-86\).