교과서 질문
A continuous random variable X is uniformly distributed with 0 ≤ X ≤ 20.
b. What is P(0 ≤ X ≤ 5)?
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6. Normal Distribution and Continuous Random Variables
Uniform Distribution
Multiple 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 =
B
No, because the area under the curve =
C
No, because probability density functions must be flat
D
Yes, because the area under the curve =
검증된 단계별 안내1
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.
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.
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.
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.
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.
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