CDF vs PDF: Understanding the Differences

This article explains the differences between the Cumulative Distribution Function (CDF) and the Probability Density Function (PDF). A random variable, at its core, is a variable whose value is a probabilistic measurement at a given time. It essentially maps a sample space to a set of real numbers.

CDF - Cumulative Distribution Function

The CDF, or Cumulative Distribution Function, of a random variable X is defined as:

Fx(x) = P(X <= x)

Properties of CDF:

• 0 <= Fx(x) <= 1
• Fx(x) is a non-decreasing function.
• lim Fx(x) = 0 (as x approaches -∞) and lim Fx(x) = 1 (as x approaches +∞)
• Fx(x) is always continuous from the right: F(x+ε) = F(x)
• P(a < X <= b) = Fx(b) - Fx(a)
• P(X=a) = Fx(a) - Fx(a’)

Key Features of CDF:

• For discrete random variables, Fx(x) is a staircase function.
• For continuous random variables, CDF is continuous.

(Refer to CCDF basics for more information.)

PDF - Probability Density Function

The PDF, or Probability Density Function, of a random variable X is defined as the derivative of the CDF:

Fx(x) = d/dx (Fx(x))

Properties of PDF:

• Fx(x) >= 0
• ∫ (from -∞ to +∞) Fx(x) dx = 1 (total probability)
• ∫ (from a+ to b-) Fx(x) dx = P(a < X <= b)
• Fx(x) = ∫ (from -∞ to x t) Fx(u) du

For discrete random variables, it’s more common to define the Probability Mass Function (PMF):

PMF = {Pi}

Where, Pi = P(X = xi)