Multiple Choice
Determine if each curve (in orange) is a valid probability density function (i.e. if the total area under the function = 1)
6. Normal Distribution & Continuous Random Variables
Uniform Distribution
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Join thousands of students who trust us to help them ace their exams!Watch the first videoMultiple Choice
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 =
C
No, because the curve does not touch the x-axis
D
Yes, because the area under the curve is slightly more than 1.
Verified step by step guidance1
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.
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.
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.
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.
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.
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