Statistical Techniques for Transportation Engineering

$\begin{array}{ll}{\mu }_4\hfill & ={\mu \prime }_4-4{\mu \prime }_3{\mu \prime }_1+6{\mu \prime }_2{({\mu \prime }_1)}^2-3{({\mu \prime }_1)}^4\hfill \\ =26–4(5.5)(1)+6(2.5)(1)+3{(1)}^4\hfill \\ =26–22+15–3=16\hfill \\ {\mu }_1\hfill & =0,{\mu }_2=1.5,{\mu }_3=0,{\mu }_4=16\hfill \end{array}$

4.17.2 Moments About Origin

Definition 4.25

The rth moment about origin of a random variable X, denoted by V_r is the expected value of X^r, 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

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

when X is continuous. ...