# Continuous random variables: Probability density functions

## Continuous random variables: Probability density functions

## Continuous Random Variables

- A
**continuous random variable**is a variable linked to the outcomes of a random event, where the variable can take on any value in a given range (interval), and not just distinct (discrete) values. - Each outcome does not have a specific probability attached to it. Instead, probabilities are determined for ranges of outcomes.

## Concept of Probability Density

- The concept of
**probability density**is used for continuous random variables. - The probability that a continuous random variable falls within a specific range is given by the area under the curve of the variable’s
**probability density function**(pdf) within that range. - The total area under a pdf always equals 1.

## Probability Density Functions (pdf)

- A
**probability density function (pdf)**is a function that describes the likelihood of different outcomes for a continuous random variable. - It is represented by
**f(x)**, where x is the possible outcome. - It is a curve such that for any two numbers a and b with a < b, the probability that the random variable takes a value between a and b equals the area under the pdf curve from a to b.
- The pdf must satisfy two conditions:

1) f(x) ≥ 0 for all x in the domain of the function.

2) The total area under the curve is 1.

## Expected Value and Variance

- The
**expected value**, or mean, of a continuous random variable is given by the integral of xp(x)dx over the whole range of x. - The
**variance**of a continuous random variable is given by the integral of (x - µ)²f(x)dx over the whole range of x, where µ is the mean. - The
**standard deviation**is the square root of the variance.

## Cumulative Distribution Function (CDF)

- In relation to continuous variables, the
**cumulative distribution function (CDF)**gives the probability that the random variable is less than or equal to the given value. It is the integral of the pdf from negative infinity to x. - All CDFs are non-decreasing functions that tend towards 1 as x → ∞.

## Application of Probability Density Function

- Probability density functions are used to model the distribution of continuous random variables. They are particularly useful in scenarios where values can vary continuously within a certain range, such as the measurement of physical quantities like height, weight, temperature, etc.