A cereal manufacturer is aware that the weight of the product in the box varies slightly from box to box. In fact, considerable historical data have allowed the determination of the density function that describes the probability structure for the weight of the product in the box (in ounces). Letting X be the random variable weight of the product in the box (in ounces), the density function can be described as: f(x) = 0.64x - 15.2, 23.75 ≤ x ≤ 25
-0.64x + 16.8, 25 ≤ x ≤ 26.25
0, elsewhere.
Important: Do not use the function's geometry in your probability calculations.
a. Verify that this is a valid density function.
b. Provide the graph for the probability density function (PDF).
c. Define the cumulative distribution function (CDF) for the random variable.
d. Determine the probability that the weight of the product in a box ranges from 23.75 to 25 ounces.
e. Determine the probability that the weight of the product in a box ranges from 25 to 26.25 ounces.
f. The company decides to classify the bottom 5% of the weights as an extremely rare occurrence. What is the maximum weight that could be classified as an extremely rare occurrence?