|
Size: 363
Comment: Initial commit
|
Size: 2125
Comment: Fix 2
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 13: | Line 13: |
| Functions can be described by singular measures that relate to their shape. For example, the '''first moment''' is the [[Calculus/AverageFunctionValue|average function value]]: | Functions can be described by singular measures that relate to their shape. For example, the '''first moment''' of a locally integrable function ''f'' that is defined between ''a'' and ''b'' is: {{attachment:first.svg}} Compare to the [[Calculus/AverageFunctionValue|average function value]] of the same function ''f'': |
| Line 16: | Line 20: |
These are similar but subtly different calculations. ---- == Random Variables == Moments are used to describe distributions of random variables. === First Moment === For a probability distribution, the first moment is the [[Analysis/ExpectedValue|expected value]]. The first moment of a continuous quantitative (numeric) random variable is expressed and notated as: {{attachment:cont1.svg}} where ''Ω'' is the sample space and ''f'' is the probability density function for that distribution. The first moment of a discrete quantitative random variable is: {{attachment:dis1.svg}} where ''Ω'' is the set of possible values and ''f'' is the probability mass function for that distribution (although the distributions of discrete random variables are commonly given as tables, rather than mathematical functions). === Second Moment === The second (raw) moment of a continuous quantitative random variable is: {{attachment:cont2.svg}} This measure isn't usually used directly however. More useful is the '''second centered moment''', which is to say the second moment around the center. {{attachment:cont2center.svg}} The most straightforward center for a distribution is the first moment itself, and by substituting that in for ''c'' above you arrive at the formal definition of [[Analysis/Variance|variance]]. === Nth Moment === Abstractly, the ''n''th moment of a continuous quantitative random variable is: {{attachment:contn.svg}} And the ''n''th moment of a discrete quantitative random variable is: {{attachment:disn.svg}} |
Moment
A moment is a description of a function's shape.
Description
Functions can be described by singular measures that relate to their shape. For example, the first moment of a locally integrable function f that is defined between a and b is:
Compare to the average function value of the same function f:
These are similar but subtly different calculations.
Random Variables
Moments are used to describe distributions of random variables.
First Moment
For a probability distribution, the first moment is the expected value.
The first moment of a continuous quantitative (numeric) random variable is expressed and notated as:
where Ω is the sample space and f is the probability density function for that distribution.
The first moment of a discrete quantitative random variable is:
where Ω is the set of possible values and f is the probability mass function for that distribution (although the distributions of discrete random variables are commonly given as tables, rather than mathematical functions).
Second Moment
The second (raw) moment of a continuous quantitative random variable is:
This measure isn't usually used directly however. More useful is the second centered moment, which is to say the second moment around the center.
The most straightforward center for a distribution is the first moment itself, and by substituting that in for c above you arrive at the formal definition of variance.
Nth Moment
Abstractly, the nth moment of a continuous quantitative random variable is:
And the nth moment of a discrete quantitative random variable is:
