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.

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.

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.

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.

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 → ∞.

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.