# Continuous random variables: Cumulative distribution functions

## Continuous Random Variables

• A continuous random variable can take any value within a specified range or interval and are measured rather than counted.
• Unlike discrete random variables, the probability that a continuous random variable equals any particular value is exactly zero. This is due to the infinite number of potential outcomes.
• For any two numbers ‘a’ and ‘b’ where a ≤ b, the probability that the continuous random variable takes a value between these two, is defined by calculating the area under a curve (often referred to as probability density function or PDF) between a and b.

## Probability Density Function (PDF)

• A probability density function of a continuous random variable is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.
• Mathematically, for a PDF of a continuous random variable ‘f(x)’, the probability that ‘X’ is in the set of outcomes ‘A’, denoted by P(X ∈ A), is defined as the area above ‘A’ and under the graph of the density function.

## Cumulative Distribution Function (CDF)

• For any continuous random variable ‘X’, the cumulative distribution function is the probability that ‘X’ will take a value less than or equal to ‘x’.
• It is defined as `F(x) = P(X ≤ x)`.
• The CDF of a continuous random variable ‘X’ is a function that is monotonically increasing, right-continuous, and defined for all real numbers.

## Properties of Cumulative Distribution Function

• The value of Cumulative Distribution Function at -∞ is equal to zero and at +∞ is equal to one.
• The CDF is always non-decreasing, or remains constant or increases.
• The CDF of a continuous random variable is a continuous function.

## Calculation of Cumulative Distribution Function

• To calculate the CDF of a continuous random variable ‘X’ at the point ‘x’, we integrate the probability density function from the left tail of the distribution to ‘x’.
• If `f(x)` is the PDF of a random variable ‘X’, then the cumulative distribution function is given by `F(x) = ∫[from -∞ to x] f(t) dt`.

## Application of Cumulative Distribution Function

• The cumulative distribution function has various practical applications, it is a fundamental tool in statistics for describing the distribution of random variables.
• It is used in hypothesis testing, construction of confidence intervals, and many other statistical procedures.
• This function can also determine the probability that a random observation that is taken from the population will be less than or equal to a certain value.