Finding the PDF from a Given Graph
When a probability density function (PDF) is given as a graph, the goal is to write its mathematical expression from the picture.
Step 1: Identify the Support
The support is the interval where the graph is above the x-axis. From the graph:
-1 ≤ x ≤ 1
Outside this interval, the PDF is zero.
Step 2: Identify Key Points
Read the important coordinates from the graph:
(-1, 0)(0, 1)(1, 0)
The graph is piecewise linear, forming a triangle.
Step 3: Find the Equation of Each Line
Left side: from -1 to 0
Slope:
m = (1 - 0) / (0 - (-1)) = 1
Equation of the line:
fX(x) = x + 1 for -1 ≤ x ≤ 0
Right side: from 0 to 1
Slope:
m = (0 - 1) / (1 - 0) = -1
Equation of the line:
fX(x) = 1 - x for 0 ≤ x ≤ 1
Step 4: Write the Piecewise PDF
fX(x) =
x + 1, -1 ≤ x ≤ 0
1 - x, 0 ≤ x ≤ 1
0, otherwise
Step 5: Verify the PDF
The total area under the curve must equal 1. The graph is a triangle with:
- Base = 2
- Height = 1
Area = (1/2) × 2 × 1 = 1
This confirms it is a valid PDF.
Key Takeaway
- Identify the interval where the PDF is nonzero
- Read key points from the graph
- Write equations for each line segment
- Combine them into a piecewise function
- Check that the total area equals 1