Central moment
In statistics central moment or focused of order k{displaystyle k} of a random variable X{displaystyle X} is mathematical hope E [chuckles](X− − E[chuckles]X])k]{displaystyle operatorname {E} [X-E[X])^{k}}}} where E{displaystyle operatorname {E} } is the operator of hope. If a random variable has no mean the central moment is indefinite. It can also be defined as:
![{displaystyle operatorname {E} [(X-E[X])^{k}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/672c58ce9f6e06feb951921062def73500be2036)
μ μ k=E [chuckles](X− − E [chuckles]X])k]=∫ ∫ − − ∞ ∞ +∞ ∞ (x− − μ μ )kf(x)dx.{displaystyle mu _{k}=operatorname {E} left[(X-operatorname {E} [X])^{k}right]=int _{-infty }^{+infty }(x-mu)^{k}f(x),dx. !
![{displaystyle mu _{k}=operatorname {E} left[(X-operatorname {E} [X])^{k}right]=int _{-infty }^{+infty }(x-mu)^{k}f(x),dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a480a097ae173d5d2a8d777447d1a6ac6bb5f2d2)
Usually the Greek letter for central moment is μ. The first central moment is zero and the second is called the variance (σ²) where σ is the standard deviation. The third and fourth central moments serve to define the standard moments called skewness and kurtosis.
The formulas for the centered and uncentered moments can be obtained from the expectation formula. If we develop it, we obtain that the uncentered moments are:
Order 0: E[chuckles]x0]=1{displaystyle E[x^{0}]=1}
![{displaystyle E[x^{0}]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf453c5af11b49f5a191c14d11ad5f8a0fb3cc3)
Order 1: E[chuckles]x]=␡ ␡ ixin=m=α α {displaystyle E[x]={frac {sum _{i}x_{i}}{n}}}{m=alpha }
![{displaystyle E[x]={frac {sum _{i}x_{i}}{n}}=m=alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0946c18d8243df7d89e3d74090bbbcdf2e394b4)
Order 2: E[chuckles]x2]=␡ ␡ ixi2n=α α 2{displaystyle E[x^{2}]={frac {sum _{i}x_{i}{i}{2}{n}}}{alpha _{2}}}}
![{displaystyle E[x^{2}]={frac {sum _{i}x_{i}^{2}}{n}}=alpha _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a2c40626498935aabc14203680296302cfc240)
Order 3: E[chuckles]x3]=␡ ␡ ixi3n=α α 3{displaystyle E[x^{3}]={frac {sum _{i}x_{i}{i}{3}{n}}}{alpha _{3}}}}
![{displaystyle E[x^{3}]={frac {sum _{i}x_{i}^{3}}{n}}=alpha _{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e19195ef195cd41d99f5466690d47bddbf989b)
Order 4: E[chuckles]x4]=␡ ␡ ixi4n=α α 4{displaystyle E[x^{4}]={frac {sum _{i}x_{i}{i}{4}{n}}}{alpha _{4}}}}
![{displaystyle E[x^{4}]={frac {sum _{i}x_{i}^{4}}{n}}=alpha _{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86b8b7ed9b7edf0be2ca8c35da7b56cd07236329)
While the centered ones are:
Order 0: E[chuckles](x− − m)0]=1{displaystyle E[(x-m)^{0}]=1}
![{displaystyle E[(x-m)^{0}]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcce73ad07566c536347f9d17dcb5b9db84f851d)
Order 1: E[chuckles](x− − m)]=α α − − α α =0{displaystyle E[(x-m)]=alpha -alpha =0}
![{displaystyle E[(x-m)]=alpha -alpha =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/091d7083f39ff4a08f238f7fc454e908440c714a)
Order 2: E[chuckles](x− − m)2]=α α 2− − α α 2=σ σ 2=μ μ 2{displaystyle E[(x-m)^{2}]=alpha _{2}-alpha ^{2}=sigma ^{2}=mu _{2}}}}}}
![{displaystyle E[(x-m)^{2}]=alpha _{2}-alpha ^{2}=sigma ^{2}=mu _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61a430659e95d03337182509a3512a1adee172d1)
Order 3: E[chuckles](x− − m)3]=α α 3− − 3α α α α 2+2α α 3=μ μ 3{displaystyle E[(x-m)^{3}]=alpha _{3}-3alpha alpha _{2}+2alpha ^{3}=mu _{3}}}}}}
![{displaystyle E[(x-m)^{3}]=alpha _{3}-3alpha alpha _{2}+2alpha ^{3}=mu _{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81b1269da56f4c097718a2fd83b16fe9322f2a10)
Order 4: E[chuckles](x− − m)4]=α α 4− − 4α α α α 3+6α α 2α α 2− − 3α α 4=μ μ 4{displaystyle E[(x-m)^{4}]=alpha _{4}-4alpha alpha _{3}+6alpha ^{2}alpha _{2}-3alpha ^{4}=mu _{4}}}}
![{displaystyle E[(x-m)^{4}]=alpha _{4}-4alpha alpha _{3}+6alpha ^{2}alpha _{2}-3alpha ^{4}=mu _{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f997830be6e8829d7aeb7407fedea8286a619072)