PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (198)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
density function (Definition)

Let $ X$ be a discrete random variable with sample space $ \{x_1,x_2,\ldots\}$. Let $ p_k$ be the probability of $ X$ taking the value $ x_k$.

The function

\begin{displaymath} f(x)= \begin{cases} p_k & \text{if }x=x_k\ 0 & \text{otherwise} \end{cases}\end{displaymath}
is called the probability function or density function.

It must hold:

$\displaystyle \sum_{j=1}^{\infty} f(x_j)=1$

If the density function for a random variable is known, we can calculate the probability of $ X$ being on certain interval:

$\displaystyle P[a<X\leq b] = \sum_{a<x_j\leq b}f(x_j) = \sum_{a<x_j\leq b}p_j.$

The definition can be extended to continuous random variables in a direct way: The probability of $ x$ being on a given interval is calculated with an integral instead of using a summation:

$\displaystyle P[a<X\leq b] = \int_a^b f(x) dx.$

For a more formal approach using measure theory, look at probability distribution function entry.



Anyone with an account can edit this entry. Please help improve it!

"density function" is owned by drini. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: distribution function, cumulative distribution function, random variable, probability distribution function, geometric distribution

Other names:  probability function, density, probabilities function
Log in to rate this entry.
(view current ratings)

Cross-references: probability distribution function, theory, measure, summation, integral, continuous random variables, interval, calculate, random variable, function, discrete random variable
There are 26 references to this entry.

This is version 9 of density function, born on 2002-09-11, modified 2005-02-21.
Object id is 3450, canonical name is DensityFunction.
Accessed 20739 times total.

Classification:
AMS MSC: 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)