Answer:
a. P(x<1.31) = 0.8132
b. P(x>1.31) = 0.1868
c. P(0.25<x<1.31) = 0.6116
d. E(x) = 0.7667
e. E(x²) = 0.8667
f. Var(x) = 0.2789
Step-by-step explanation:
Given
f(x) = 0.85 - 0.35x 0 < x < 2
a. P(x<1.31).
This means we're working with interval of 0 to 1.31 minutes translating to Rainees completing the task between the interval 0 to 1.31 minutes
This is calculated as follows;
P(x<1.31) = ∫ f(x)dx {0,1.31}
P(x<1.31) = ∫ (0.85 - 0.35x)dx {0,1.31}
P(x<1.31) = (0.85x - 0.35x²/2){0,1.31}
P(x<1.31) = (0.85(1.31) - 0.35(1.31)²/2)
P(x<1.31) = 0.8131825
P(x<1.31) = 0.8132 ---- Approximated
b) P(x>1.31)
This means we're working with interval of 1.31 to 2 minutes translating to Rainees completing the task between the interval 1.31 to 2 minutes
This is calculated as follows;
P(x>1.31) = ∫ f(x)dx {1.31,2}
P(x>1.31) = ∫ (0.85 - 0.35x)dx {1.31,2}
P(x>1.31) = (0.85x - 0.35x²/2) {1.31,2}
P(x>1.31) = (0.85(2) - 0.35(2)²/2) - (0.85(1.31) - 0.35(1.31)²/2)
P(x>1.31) = 0.1868175
P(x>.1.31) = 0.1868 ---- Approximated
Alternatively, it can be calculated as;
P(x>1.31) = 1 - P(x<1.31)
P(x>1.31) = 1 - 0.8132
P(x>1.31) = 0.1868
c) This means we're working with interval of 0.25 to 1.31 minutes translating to Rainees completing the task between the interval 0.25 to 1.31 minutes.
This is calculated as follows
P(0.25<x<1.31) = ∫ f(x)dx {0.25,1.31}
P(0.25<x<1.31) = ∫ (0.85 - 0.35x)dx {0.25,1.31}
P(0.25<x<1.31) = (0.85x - 0.35x²/2) {0.25,1.31}
P(0.25<x<1.31) = (0.85(1.31) - 0.35(1.31)²/2) - (0.85(0.25) - 0.35(0.25)²/2)
P(0.25<x<1.31) = 0.61162
P(0.25<x<1.31) = 0.6116 ---- Approximated
d) Expected time, E(x) is calculated as
E(x) = ∫xf(x) dx
E(x) =∫ x(0.85 - 0.35x)dx {0,2}
E(x) =∫ (0.85x - 0.35x²)dx {0,2}
E(x) =(0.85x²/2 - 0.35x³/3) {0,2}
E(x) =(0.85(2)²/2 - 0.35(2)³/3)
E(x) = 0.766666666666666
E(x) = 0.7667 --- Approximated
e)
E(x²) is calculated as;
E(x²) = ∫x²f(x) dx
E(x²) =∫ x²(0.85 - 0.35x)dx {0,2}
E(x²) =∫ (0.85x² - 0.35x³)dx {0,2}
E(x²) =(0.85x³/3 - 0.35x⁴/4) {0,2}
E(x²) = (0.85(2)³/3 - 0.35(2)⁴/4)
E(x²) = 0.866666666666666
E(x²) = 0.8667 ---- Approximated
f) Var(x) = E(x²) - (E(x))²
Var(x) = 0.8667 - (0.7667)²
Var(x) = 0.27887111
Var(x) = 0.2789 --- Approximated