In probability and statistics, several terms are used to describe the various functions that are used to model probability distributions. These include:

**Probability density function (PDF):**

The PDF is a function that describes the probability of a **continuous** random variable **taking on a certain value**. It is a mathematical function that describes the probability that a random variable will fall within a certain range of values.

**Cumulative distribution function (CDF): **

The CDF is a function that describes the probability that a random variable (**continuous or discrete**) will **take on a value less than or equal to a certain value**. It is a mathematical function that describes the probability that a random variable will fall within a certain range of values, up to and including a specific value.

**Probability mass function (PMF): **

The PMF is a function that describes the probability of a **discrete** random variable **taking on a certain value**. It is a mathematical function that describes the probability that a random variable will take on a specific value rather than falling within a range of values.

## Summary:

In summary, PDFs are used to describe the probability of a continuous random variable taking on a certain value, CDFs are used to describe the probability that a random variable (continuous or discrete) will take on a value less than or equal to a certain value, and PMFs are used to describe the probability of a discrete random variable taking on a certain value.

Feature | Probability Density Function (PDF) | Cumulative Distribution Function (CDF) | Probability Mass Function (PMF) |
---|---|---|---|

Function of | Describes the likelihood of a continuous random variable taking on a specific value or falling within a specific interval | Describes the probability that a random variable (continuous or discrete) will take on a value less than or equal to a certain value | Describes the probability of a discrete random variable taking on a specific value |

Types of Variables | Continuous random variables | Continuous and discrete random variables | Discrete random variables |

Probability | The value of the PDF at a specific point indicates the probability density, not the probability itself | The value of the CDF at a specific point indicates the probability that the random variable will take on a value less than or equal to that point | The value of the PMF at a specific point indicates the probability of the random variable taking on that value |

Sum or Integral | The integral of the PDF over the entire range of the random variable equals 1 | The limit of the CDF as the random variable approaches positive infinity is 1 | The sum of the PMF over all possible values of the random variable equals 1 |

Visualization | The PDF can be visualized as a curve, with the area under the curve representing the probability | The CDF can be visualized as a curve that starts at 0 and increases monotonically to 1 | The PMF can be visualized as a bar chart, with each bar representing the probability of a specific value |