What are moments in probability theory
Moments of distribution functions
Moments are parameters of a distribution function of a random variable. They correspond to the parameters of descriptive statistics. The terms expected value, variance, skewness and curvature for describing a function result from the so-called central moments (see there). A distribution function is determined by specifying all of its moments, if they exist. There are also distributions whose moments do not exist, such as B. the Lévy distribution. One distinguishes ordinary moments, absolute, central and the Moment at c.
Example: A normal distribution is determined, for example, by its expected value and its second moment, since all odd-numbered moments vanish and the higher even-numbered moments are directly related to the second moment.
Definition - Ordinary Moments
Let X be a random variable, k a natural and r a real number. Then it is referred to as ordinary moment of order k with respect to r (or simply as the kth ordinary moment) the expected value of the kth power of the random variable "centered" on r:
- mk (r) = E ((X − r) k)
Continuous random variable
With a continuous real random variable with the probability density function fX we get:
- mk (r): = - ∞∫∞ (x − r) kfX (x) dx
Discrete random variable
- mk (r) = i = 1∑∞ (xi −r) kpi
ordinary moments (order k)
- m1 = E (x)
- m2 = E (x2) = Var (x) + (E (x)) 2
- Mk (r) = E (∣ (X − r) ∣k)
is referred to as k th absolute moment of x with respect to r.
The central moments substitute the expected value E (X) itself for r.
- μk: = E ((X − m1) k)
The central momentfirst order is equal to 0.
- μ1 = 0
The central momentsecond order corresponds to the variance.
- μ2 (μ) = E ((X − m1) 2)
The central momentthird order corresponds to the skew with γ ∗ σ3.
Moment and the characteristic function
By deriving the formula for the characteristic function several times, one obtains a representation of the ordinary moments through the characteristic function as:
- E (Xk) = ikφX (k) (0) (k = 1,2, ...)
Moment around a constant (c)
- the moment around c (c: constant, kth order): E (x − c) k
Moments around zero
If r = 0, one speaks of Moments around zero, or designated
- mk = mk (0) = E ((X − 0) k) = E (Xk)
simply as the kth moment. The kth moment can be determined with the moment generating function.
Mathematics as a subject is so serious that no opportunity should be missed to make it more entertaining.
Provider identification: Mathеpеdιa von Тhοmas Stеιnfеld • Dοrfplatz 25 • 17237 Blankеnsее • Tel .: 01734332309 (Vodafone / D2) • Email: cο@maτhepedιa.dе
- Are twins spiritually connected
- Do you wear shorts?
- The old Romans had glass
- Will society collapse due to climate change?
- How do I find a niche website
- How can you get banned from Walmart
- Who created the minimum wage
- How do I eliminate clickbait on Facebook
- What is the best weapon with a blade
- Which villain actually had a point
- Why is the value of money falling?
- How does masturbation affect our health?
- How can we reduce the drinking of minors?
- What is the age limit for skydiving
- Is it possible to be 100 vegetarian?
- Why are airliners put under pressure
- Is moss organic or inorganic
- Is feminism misunderstood why or why not
- Polyamory what should we do
- How do fish see at night
- What are the most frequently asked Java questions
- What existed before existence
- How much do kestrels cost
- How can I sell my OneCoin