Fundamental Theorem of Calculus and Its Relationship with CDFs

1. Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculus connects differentiation and integration. It has two parts:

1. Part 1: If F(x) is the antiderivative of f(x), then:

F'(x) = f(x)

2. Part 2: The definite integral of f(x) from a to b is given by:

∫[a to b] f(x) dx = F(b) − F(a)

These relationships show that integration and differentiation are inverse operations.

2. Relationship Between FTC and CDFs

The Fundamental Theorem of Calculus directly applies to cumulative distribution functions (CDFs) and probability density functions (PDFs). Here's how:

• The Cumulative Distribution Function (CDF), F(x), describes the probability that a random variable X is less than or equal to x:

F(x) = P(X ≤ x) = ∫[-∞ to x] f(t) dt

• The Probability Density Function (PDF), f(x), represents the rate of change of the CDF:

f(x) = F'(x)

3. Intuition: Connecting Area and Probability

The CDF accumulates probability up to a certain value x, analogous to finding the area under a curve. The PDF, being the derivative of the CDF, tells us how "fast" the probability is accumulating at x.

For example:

• If the PDF f(x) is high at a point, the CDF F(x) increases rapidly around that point.
• If the PDF is low, the CDF grows more slowly.

4. Example: Standard Normal Distribution

Consider the standard normal distribution:

• PDF: f(x) = (1 / √(2π)) e^(-x² / 2)
• CDF: F(x) = ∫[-∞ to x] f(t) dt

The CDF at x = 0 gives the probability that X is less than or equal to 0, which is 0.5. The PDF at x = 0 is the maximum value of the density, reflecting the peak of the normal curve.

5. Applications of the Relationship

The connection between CDFs and PDFs has many applications:

• Statistics: Computing probabilities, quantiles, and expectations.
• Machine Learning: Likelihood estimation and generative models.
• Finance: Modeling asset prices and risk measures.

6. Key Insights

• The CDF F(x) represents cumulative probability, and the PDF f(x) represents the density of the random variable.
• By the Fundamental Theorem of Calculus, the PDF is the derivative of the CDF, and the CDF is the integral of the PDF.
• This duality underpins much of probability theory and its practical applications.

Exercise