Calculate CDF from PDF

Understanding the Relationship Between PDF and CDF

At its core, the CDF is just the accumulated area under the PDF.

Definitions

Let X be a continuous random variable.

• PDF: fX(x)
• CDF: FX(x) = P(X ≤ x)

Mathematical Relationship

The CDF is the integral of the PDF:

FX(x) = ∫-∞x fX(t) dt
      

Step-by-Step Method

1. Identify the support of the PDF
   Determine where fX(x) is nonzero.
2. Integrate the PDF
   This often gives:

   FX(x) =
   0,              x < a
   ∫ax fX(t) dt,   a ≤ x ≤ b
   1,              x > b
             

3. Check endpoints
   • FX(-∞) = 0
   • FX(∞) = 1

Example

Given the PDF:

fX(x) =
2x,   0 ≤ x ≤ 1
0,    otherwise
      

Compute the CDF:

For x < 0:

FX(x) = 0

For 0 ≤ x ≤ 1:

FX(x) = ∫0x 2t dt = x2
      

For x > 1:

FX(x) = 1

Final CDF

FX(x) =
0,      x < 0
x²,     0 ≤ x ≤ 1
1,      x > 1
      

Intuition

• PDF → height
• CDF → accumulated area
• Differentiating the CDF returns the PDF:
  fX(x) = dFX(x)/dx