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.

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.

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.

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.

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.

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.