# Cumulative distribution functions

## Cumulative distribution functions

Understanding the Cumulative Distribution Function (CDF)

• The Cumulative Distribution Function, often abbreviated as CDF, is applied to continuous random variables. It gives the cumulative probability from negative infinity up to a given value.
• For a continuous random variable X, the CDF at point x is denoted as F(x) = P(X ≤ x). This means that the CDF gives the probability that the random variable is less than or equal to a certain value.
• The CDF is calculated as the integral of the probability density function (pdf) from negative infinity to the given point.
• The CDF is always between 0 and 1. The closer the value of the CDF is to one, then the greater the likelihood that a randomly selected value from the distribution will fall within that range.
• The CDF graph is monotonic, which means it never decreases as it moves along the x-axis. This is because it’s impossible for the cumulative probability to decrease as we move up along the range of possible values.

Properties of the CDF

• The function of the CDF begins at 0 and increases or remains steady up to a value of 1.
• For any two distinct values a and b where a is less than b, F(a) will always be less than or equal to F(b). This is due to monotonous nature of the CDF.
• The value of the CDF at negative infinity is 0, and at positive infinity is 1.

CDF and Probability Intervals

• The probability that a continuous random variable falls within a certain interval can be calculated using the CDF. This is given by the formula P(a ≤ X ≤ b) = F(b) - F(a).
• Using the CDF to find the probability over an interval is handy as it eliminates the need to compute definite integrals for each calculation.

Inverse Cumulative Distribution Function

• The inverse CDF, also known as the Quantile Function, outputs the value for which the cumulative distribution of a random variable is less than or equal to a given probability.

Remember, beyond memorising the respective formulas, understanding the underlying concepts and logic is crucial! Be sure to revise your Integration techniques as it forms the fundamental tool for calculating CDF from pdf.