# moment

Moments

Given a random variable^{} $X$, the $k$th moment of $X$ is the value $E[{X}^{k}]$, if the expectation exists.

Note that the expected value is the first moment of a random variable, and the variance^{} is the second moment minus the first moment squared.

The $k$th moment of $X$ is usually obtained by using the moment generating function.

Given a random variable $X$, the $k$th central moment of $X$ is the value $E\left[{(X-E[X])}^{k}\right]$, if the expectation exists. It is denoted by ${\mu}_{k}$.

Note that the ${\mu}_{1}=0$ and ${\mu}_{2}=Var[X]={\sigma}^{2}$. The third central moment divided by the standard deviation^{} cubed is called the *skewness* $\tau $:

$$\tau =\frac{{\mu}_{3}}{{\sigma}^{3}}$$ |

The skewness measures how “symmetrical”, or rather, how “skewed”, a distribution^{} is with respect to its mode. A non-zero $\tau $ means there is some degree of skewness in the distribution. For example, $\tau >0$ means that the distribution has a longer positive^{} tail.

The fourth central moment divided by the fourth power of the standard deviation is called the *kurtosis* $\kappa $:

$$\kappa =\frac{{\mu}_{4}}{{\sigma}^{4}}$$ |

The kurtosis measures how “peaked” a distribution is compared to the standard normal distribution^{}. The standard normal distribution has $\kappa =3$. $$ means that the distribution is “flatter” than then standard normal distribution, or *platykurtic*. On the other hand, a distribution with $\kappa >3$ can be characterized as being more “peaked” than $N(0,1)$, or *leptokurtic*.

Title | moment |

Canonical name | Moment |

Date of creation | 2013-03-22 11:53:54 |

Last modified on | 2013-03-22 11:53:54 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 11 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 60-00 |

Classification | msc 62-00 |

Classification | msc 81-00 |

Defines | central moment |

Defines | skewness |

Defines | kurtosis |

Defines | platykurtic |

Defines | leptokurtic |