# Cumulative Distribution Functions and Probability Density Functions

## Cumulative Distribution Function (CDF)

**The Cumulative Distribution Function (CDF)** is a function that describes the cumulative probability of a random variable taking on a value less than or equal to a given value.

Mathematically, the CDF of a random variable X, denoted as F(x), is defined as:

`F(x) = Probability that variable X is less or equal to value x`

.

Using this function, it is easy to describe continuous random variables.

Look at the example below: we will use a normally distributed random variable and look at its CDF using the `.cdf()`

method.

Using CDF, we can determine the probability that our random variable belongs to any of the intervals of interest. Assume that X is a random variable, and F(x) is its CDF.**To determine the probability that the variable X belongs to the interval [a, b], we can use the following formula**:

`P{X є [a,b]} = F(b) - F(a)`

.

## Percent Point Function (PPF)

**Percent Point Function (PPF)**, also known as the inverse of the cumulative distribution function (CDF). It is used to find the value of a random variable that **corresponds to a given probability**. In Python it is implemented using `.ppf()`

method:

## Probability Density Function (PDF)

**Probability Density Function (PDF)** is a function that provides information about the likelihood of a random variable taking on a particular value at a specific point in the continuous range. Its interpretation is similar to that of the PMF but is specifically used for describing continuous random variables.

The PDF defines the **shape of the probability distribution** of a continuous random variable.

Let's consider the following example of PDF calculated using the `.pdf()`

method.

The PDF provides insight into the **likelihood** or **probability density** of a random variable assuming a specific value. Higher PDF values suggest a greater likelihood, while lower values suggest a lesser likelihood.

To determine the probability of a continuous variable falling within a specific range, similar to using the PMF, we calculate the sum of the PDF for all values within that range. However, since continuous variables can have an infinite number of values within any range, we calculate the **area under the PDF curve** within the specified range instead of a simple sum.

Everything was clear?

Course Content

Probability Theory Mastering

## Probability Theory Mastering

1. Additional Statements From The Probability Theory

3. Estimation of Population Parameters

4. Testing of Statistical Hypotheses

# Cumulative Distribution Functions and Probability Density Functions

## Cumulative Distribution Function (CDF)

**The Cumulative Distribution Function (CDF)** is a function that describes the cumulative probability of a random variable taking on a value less than or equal to a given value.

Mathematically, the CDF of a random variable X, denoted as F(x), is defined as:

`F(x) = Probability that variable X is less or equal to value x`

.

Using this function, it is easy to describe continuous random variables.

Look at the example below: we will use a normally distributed random variable and look at its CDF using the `.cdf()`

method.

Using CDF, we can determine the probability that our random variable belongs to any of the intervals of interest. Assume that X is a random variable, and F(x) is its CDF.**To determine the probability that the variable X belongs to the interval [a, b], we can use the following formula**:

`P{X є [a,b]} = F(b) - F(a)`

.

## Percent Point Function (PPF)

**Percent Point Function (PPF)**, also known as the inverse of the cumulative distribution function (CDF). It is used to find the value of a random variable that **corresponds to a given probability**. In Python it is implemented using `.ppf()`

method:

## Probability Density Function (PDF)

**Probability Density Function (PDF)** is a function that provides information about the likelihood of a random variable taking on a particular value at a specific point in the continuous range. Its interpretation is similar to that of the PMF but is specifically used for describing continuous random variables.

The PDF defines the **shape of the probability distribution** of a continuous random variable.

Let's consider the following example of PDF calculated using the `.pdf()`

method.

The PDF provides insight into the **likelihood** or **probability density** of a random variable assuming a specific value. Higher PDF values suggest a greater likelihood, while lower values suggest a lesser likelihood.

To determine the probability of a continuous variable falling within a specific range, similar to using the PMF, we calculate the sum of the PDF for all values within that range. However, since continuous variables can have an infinite number of values within any range, we calculate the **area under the PDF curve** within the specified range instead of a simple sum.

Everything was clear?