Determine if each curve (in orange) is a valid probability density function (i.e. if the total area under the function = 1)

• A. Yes, because the area under the curve = $1$
• B. No, because the area under the curve = $2\ne1$
• C. No, because probability density functions must be flat
• D. Yes, because the area under the curve = $2\thickapprox1$

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the definition of a probability density function (PDF). A PDF must satisfy two conditions: (1) The function must be non-negative for all values of x, and (2) The total area under the curve must equal 1.
2. Step 2: Analyze the graph provided. The orange curve starts at (2.5, 0), increases linearly to (10, 0.25), and then becomes flat from (10, 0.25) to (12.5, 0). This indicates the function is piecewise linear.
3. Step 3: Break the area under the curve into two regions: (1) The triangular region from x = 2.5 to x = 10, and (2) The rectangular region from x = 10 to x = 12.5. Calculate the area of each region separately.
4. Step 4: Use the formula for the area of a triangle, A = 0.5 * base * height, to calculate the area of the triangular region. The base is (10 - 2.5) and the height is 0.25. For the rectangular region, use the formula A = base * height, where the base is (12.5 - 10) and the height is 0.25.
5. Step 5: Add the areas of the two regions together. If the total area equals 1, the orange curve is a valid probability density function. If the total area is not equal to 1, it is not a valid PDF.