Moments about arbitrary constant,their relation with moments about mean: measures of dispersion p.11 скачать в хорошем качестве

Moments about arbitrary constant,their relation with moments about mean: measures of dispersion p.11

Moments about arbitrary constant Measures of Dispersion Part 11 Moment about Arbitrary constant Arbitrary constant means any constant of our choice say a. The population moment about arbitrary constants are denoted by (𝝁_𝟏 ) ́,(𝝁_𝟐 ) ́ 𝒆𝒕𝒄. (𝝁_𝒓 ) ́=𝟏/𝑵∑(𝒙_𝒊−𝒂)^𝒓,𝒓=𝟏,𝟐,𝟑,𝟒,−,−,− Moment about Arbitrary constant The sample moment about arbitrary constants are denoted by (𝒎_𝟏 ) ́,(𝒎_𝟐 ) ́ 𝒆𝒕𝒄. (𝒎_𝒓 ) ́=𝟏/𝒏∑(𝒙_𝒊−𝒂)^𝒓,𝒓=𝟏,𝟐,𝟑,𝟒,−,−,− Moments about zero The rth moment about zero is given as (𝒎_𝒓 ) ́=𝟏/𝒏∑𝒙^𝒓, 𝒓=𝟏,𝟐,𝟑,𝟒,−,−,− Moment about mean in terms of moments about arbitrary constant The following relationship exist between moments about mean and moment about arbitrary constant, 𝒎_𝟏=(𝒎_𝟏 ) ́−(𝒎_𝟏 ) ́=𝟎 𝒎_𝟐=(𝒎_𝟐 ) ́ −((𝒎_𝟏 ) ́ )^𝟐 𝒎_𝟑=(𝒎_𝟑 ) ́−𝟑(𝒎_𝟐 ) ́(𝒎_𝟏 ) ́+𝟐((𝒎_𝟏 ) ́ )^𝟑 𝒎_𝟒=(𝒎_𝟒 ) ́−𝟒(𝒎_𝟑 ) ́(𝒎_𝟏 ) ́+𝟔(𝒎_𝟐 ) ́((𝒎_𝟏 ) ́ )^𝟐−𝟑((𝒎_𝟏 ) ́ )^𝟒 Finding mean from first arbitrary moment The mean can be obtained using the relationship 𝑿 ̅=𝒂+(𝒎_𝟏 ) ́