# density function

Let $X$ be a discrete random variable with sample space $\{{x}_{1},{x}_{2},\mathrm{\dots}\}$. Let ${p}_{k}$ be the probability of $X$ taking the value ${x}_{k}$.

The function^{}

$$f(x)=\{\begin{array}{cc}{p}_{k}\hfill & \text{if}x={x}_{k}\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$ |

is called the *probability function ^{}* or

*density function*.

It must hold:

$$\sum _{j=1}^{\mathrm{\infty}}f({x}_{j})=1$$ |

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

$$ |

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:

$$ |

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

Title | density function |

Canonical name | DensityFunction |

Date of creation | 2013-03-22 13:02:49 |

Last modified on | 2013-03-22 13:02:49 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 12 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 60E05 |

Synonym | probability function |

Synonym | density |

Synonym | probabilities function |

Related topic | DistributionFunction |

Related topic | CumulativeDistributionFunction |

Related topic | RandomVariable |

Related topic | Distribution^{} |

Related topic | GeometricDistribution2 |