Respuesta :
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
Answer:
a) P(0 < x < 1.31) = 0.8132
b) P(1.31 < x < 2) = 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:
Probability density function = f(x) = 0.85 - 0.35x for 0 < x < 2
Note that the values obtained here are the closest mathematical estimates. A greater/less than or equal to sign would make it all more accurate.
a) probability a trainee will complete the task in less than 1.31 minutes
P(0 < x < 1.31) = ∫¹•³¹₀ f(x) dx
= ∫¹•³¹₀ (0.85 - 0.35x) dx
= [0.85x - 0.175x²]¹•³¹₀
= [(0.85)(1.31) - (0.175)(1.31²)]
= 1.1135 - 0.3003175
= 0.8131825 = 0.8132 to 4 d.p.
b) probability that a trainee will complete the task in more than 1.31 minutes
P(1.31 < x < 2) = ∫²₁.₃₁ f(x) dx
= ∫²₁.₃₁ (0.85 - 0.35x) dx
= [0.85x - 0.175x²]²₁.₃₁
= [(0.85)(2) - (0.175)(2²)] - [(0.85)(1.31) - (0.175)(1.31²)]
= (1.7 - 0.7) - (1.1135 - 0.3003175) = 0.1868175 = 0.1868
c) probability it will take a trainee between 0.25 minutes and 1.31 minutes to complete the task
P(0.25 < x < 1.31) = ∫¹•³¹₀.₂₅ f(x) dx
= ∫¹•³¹₀.₂₅ (0.85 - 0.35x) dx
= [0.85x - 0.175x²]¹•³¹₀.₂₅
= [(0.85)(1.31) - (0.175)(1.31²)] - [(0.85)(0.25) - (0.175)(0.25²)]
= 0.8131825 - 0.2015625 = 0.61162 = 0.6116
d) Expected value = ∫ xf(x) dx [with the integral done all over the sample spaces, for this question, it is done over the interval (0,2)]
E(X) = ∫²₀ x(0.85 - 0.35x) dx
= ∫²₀ (0.85x - 0.35x²) dx
= [0.425x² - 0.1167x³]²₀
= [(0.425)(2²) - (0.1167)(2³)]
= [1.7 - 0.9333333] = 0.7667
e) X represents the time it takes to complete the task, what is E(X²)
E(X²) = ∫ x²f(x) dx [with the integral done all over the sample spaces, for this question, it is done over the interval (0,2)]
E(X) = ∫²₀ x²(0.85 - 0.35x) dx
= ∫²₀ (0.85x² - 0.35x³) dx
= [0.2833x³ - 0.0875x⁴]²₀
= [(0.2833)(2³) - (0.0875)(2⁴)]
= [2.2667 - 1.4] = 0.8667
f) Variance of continuous function is given as
Var(X) = E(X²) - [E(X)]²
Var(X) = 0.8667 - 0.7667² = 0.2789
Hope this Helps!!!
