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If the first four moments of a distribution about the value 4 of the random variable are $-1.5, 1.7, -30$ and $108$ then find first four raw moments.

Mumbai University > COMPS > Sem 4 > Applied Mathematics 4

Marks : 04

Year : MAY 2014

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Given $a=4$

By definition,

Raw Moments about ‘a’ $\mu_r=E[(x-a)^{\mu}] \\ \therefore \mu_r=E[(x-4)^{\mu}] ----- (1)$

and

Raw moments about origin: $\mu_r=E[x^{\mu}]$

Put $\mu =1$ in (1), $\mu_1^1=E[(x-4)]$

$$\therefore -1.5=E(x)-4 \\ E(x)=2.5 ----------------(2)$$

Put $\mu =2$ in (1), $\mu_2^1=E[(x-4)^2]\therefore 17=E(x^2-8x+16)\\ \therefore 17=E(x^2)-8E(x)+16\\ 17=E(x^2)-8\times 2.15+16 ----- from (2)\\ E(x^2)=17+20-16\\ \therefore E(x^2) =21 ------ (3)$

Put $\mu=3$ in (1) $\mu_3^1=E[(x-4)^3]\\ \therefore 108=E(\space ^4C_0x^4-^4C_1\times x^3 \times4 + ^4C_2 \times x^2 \times 4- ^4C_2 \times x^2 \times 4^2 - ^4C_3 \times x \times 4^3+ ^4C_4 \times 4^4) \\ \therefore 108 = E(x^4) -16E(x^3) - 96 E(x^2)-256E(x) +256 \\ \therefore 108=E(x^4)-16 \times166+96 \times 21 -256\times2.5 + 256... (from \space 2,3,4)\\ \therefore E(x^4)=108+16\times166-96\times21+256\times2.5-256\\ \therefore E(x^4)=1132$

Hence, four moments about origin are $2.5,21,166,1132.$

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