The first four moments of a distribution about the origin are -1.5, 17, -30 and 108. The third moment about the mean is:
39.75
The third central moment in terms of moments about the origin is:
\[\mu_3=\nu_3-3\nu_2\nu_1+2\nu_1^{3}\]
Step 1 — Substitute \(\nu_1=-1.5,\ \nu_2=17,\ \nu_3=-30\):
\[\mu_3=-30-3(17)(-1.5)+2(-1.5)^{3}\]
Step 2 — Evaluate each term:
\[3(17)(-1.5)=-76.5,\quad 2(-1.5)^{3}=2(-3.375)=-6.75\]
Step 3 — Combine:
\[\mu_3=-30+76.5-6.75=39.75\]
Hence the third moment about the mean is 39.75.
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If μ'r and μr, respectively, denote rth order moments about origin and mean, then:
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