This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the evaluation of the probability: To evaluate the probability P(0.5 X 1.5), we use the Cumulative Distribution Function (CDF) F(x) = (3x^2)/(4) - (x^3)/(4). The probability is given by P(0.5 X 1.5) = F(1.5) - F(0.5). Step 1: Calculate F(1.5). Substitute x = 1.5 (or (3)/(2)) into the CDF: F(1.5) = (3(1.5)^2)/(4) - ((1.5)^3)/(4) F(1.5) = (3(3)/(2))^24 - ((3)/(2))^34 F(1.5) = (3(9)/(4))4 - (27)/(8)4 F(1.5) = (27)/(4)4 - (27)/(32) F(1.5) = (27)/(16) - (27)/(32) To subtract, find a common denominator, which is 32: F(1.5) = (27 × 2)/(16 × 2) - (27)/(32) = (54)/(32) - (27)/(32) = (27)/(32) Step 2: Calculate F(0.5). Substitute x = 0.5 (or (1)/(2)) into the CDF: F(0.5) = (3(0.5)^2)/(4) - ((0.5)^3)/(4) F(0.5) = (3(1)/(2))^24 - ((1)/(2))^34 F(0.5) = (3(1)/(4))4 - (1)/(8)4 F(0.5) = (3)/(4)4 - (1)/(32) F(0.5) = (3)/(16) - (1)/(32) To subtract, find a common denominator, which is 32: F(0.5) = (3 × 2)/(16 × 2) - (1)/(32) = (6)/(32) - (1)/(32) = (5)/(32) Step 3: Calculate P(0.5 X 1.5). P(0.5 X 1.5) = F(1.5) - F(0.5) P(0.5 X 1.5) = (27)/(32) - (5)/(32) P(0.5 X 1.5) = (27 - 5)/(32) = (22)/(32) Step 4: Simplify the fraction: P(0.5 X 1.5) = (11)/(16) The probability is (11)/(16). That's 2 down. 3 left today — send the next one.