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$\begin{array}{ll}{\mu }_{4}\hfill & ={\mu \prime }_{4}-4{\mu \prime }_{3}{\mu \prime }_{1}+6{\mu \prime }_{2}{\left({\mu \prime }_{1}\right)}^{2}-3{\left({\mu \prime }_{1}\right)}^{4}\hfill \\ =26–4\left(5.5\right)\left(1\right)+6\left(2.5\right)\left(1\right)+3{\left(1\right)}^{4}\hfill \\ =26–22+15–3=16\hfill \\ {\mu }_{1}\hfill & =0,{\mu }_{2}=1.5,{\mu }_{3}=0,{\mu }_{4}=16\hfill \end{array}$

Definition 4.25

The rth moment about origin of a random variable X, denoted by Vr is the expected value of Xr, symbolically

${V}_{r}=E\left[{X}^{r}\right]=\sum _{i=1}^{n}{x}_{i}^{r}{p}_{i}$

For r=0, 1, 2, … when X is discrete, and

${V}_{r}=E\left[{X}^{r}\right]={\int }_{-\infty }^{\infty }{x}^{r}f\left(x\right)\mathrm{d}x$

when X is continuous. ...

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