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 = $2\ne 1$
• C. No, because probability density functions must be flat
• D. Yes, because the area under the curve = $2\approx 1$

Answer: (A)

Verified step by step guidance

1. Step 1: Recall that a probability density function (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 consists of two segments: a sloped line from x = 2.5 to x = 10, and a flat line from x = 10 to x = 12.5.
3. Step 3: Calculate the area under the sloped line. The sloped line forms a triangle with a base of (10 - 2.5 = 7.5) and a height of 0.25. Use the formula for the area of a triangle: A = (1/2) * base * height.
4. Step 4: Calculate the area under the flat line. The flat line forms a rectangle with a base of (12.5 - 10 = 2.5) and a height of 0.2. Use the formula for the area of a rectangle: A = base * height.
5. Step 5: Add the areas of the triangle and rectangle together to determine the total area under the curve. Compare this total to 1 to verify if the curve is a valid probability density function.