The first three moments of a distribution about the value 2 are 1,16,-40.
Calculate moment about mean, moment about origin

Solution:
The first three moments of a distribution about the value 2 are
`M_1=1`

`M_2=16`

`M_3=-40`

Find Central moments using Moments about the value 2

First Central Moment
`m_1=0`

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Second Central Moment
`m_2=M_2-M_1^2`

`=16-1^2`

`=16-1`

`=15`

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Third Central Moment
`m_3=M_3-3M_2M_1+2M_1^3`

`=(-40)-3*16*1+2*1^3`

`=(-40)-48+2`

`=-86`

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Now, find `bar x`

`M_1=bar x-A`

`1=bar x-2`

`bar x=1+2`

`bar x=3`

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Find Moment about origin using Central moments

First Moment about origin
`v_1=bar x`

`v_1=3`

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Second Moment about origin
`v_2=m_2+v_1^2`

`=15+3^2`

`=15+9`

`=24`

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Third Moment about origin
`v_3=m_3+3m_2v_1+v_1^3`

`=(-86)+3*15*3+3^3`

`=(-86)+135+27`

`=76`