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 equals 1
• B. No, because the area under the curve = $0.8\ne 1$
• C. No, because the curve does not touch the x-axis
• D. Yes, because the area under the curve is slightly more than 1.

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the definition of a probability density function (PDF). A valid 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 is a horizontal line at y = 0.2 between x = 1 and x = 5. Outside this interval, the curve touches the x-axis, meaning the function is zero.
3. Step 3: Calculate the area under the curve. Since the curve is constant at y = 0.2 over the interval [1, 5], the area can be calculated using the formula for the area of a rectangle: Area = height × width. Here, height = 0.2 and width = 5 - 1 = 4.
4. Step 4: Verify if the total area equals 1. Substitute the values into the formula: Area = 0.2 × 4. Check if this result equals 1 to determine if the curve is a valid PDF.
5. Step 5: Conclude based on the calculation. If the area equals 1, the curve is a valid PDF. If the area does not equal 1, the curve is not a valid PDF.