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: Recall 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 (y = 0.2) over the interval [1, 5], the area can be calculated using the formula for the area of a rectangle: \( \text{Area} = \text{Height} \times \text{Width} \). Here, the height is 0.2 and the width is \( 5 - 1 = 4 \).
4. Step 4: Multiply the height (0.2) by the width (4) to find the total area under the curve. This will determine whether the total area equals 1, which is required for the function to be a valid PDF.
5. Step 5: Compare the calculated area to 1. If the area equals 1, the curve is a valid PDF. If the area is not equal to 1, the curve is not a valid PDF.