Trainees must complete a specific task in less than 2 minutes. Consider the prob
ID: 3048824 • Letter: T
Question
Trainees must complete a specific task in less than 2 minutes. Consider the probability density function below for the time it takes a trainee to complete the task.
f(x) = 0.76 - 0.26x
0 < x < 2
a) What is the probability a trainee will complete the task in less than 1.26 minutes? Give your answer to four decimal places.
b) What is the probability that a trainee will complete the task in more than 1.26 minutes? Give your answer to four decimal places.
c) What is the probability it will take a trainee between 0.37 minutes and 1.26 minutes to complete the task? Give your answer to four decimal places.
d) What is the expected time it will take a trainee to complete the task? Give your answer to four decimal places.
e) If X represents the time it takes to complete the task, what is E(X2)? Give your answer to four decimal places.
f) If X represents the time it takes to complete the task, what is Var(X)? Give your answer to four decimal places.
Explanation / Answer
1. The probability will be the area of the trapezoid formed under the curve and bounded below by the x-axis. The bases of the trapezoid are:
f(0) = 0.76 and f(1.26) = 0.4324
The height of the trapezoid is 1.26.
The area is A = (1/2)(0.76 + 0.4324)(1.26) = .7512
2. 1 - .7512 = .2488
3. Again, find the area of the trapezoid...
The bases are f(.37) = .6638 and f(1.26) = 0.4324
The height of the trapezoid is 1.26 - .37 = .89
The area is: A = (1/2)(.6638 + 0.4324)(.89) = .4878
4. E(X) = integral (from 0 to 2) of x(0.76 - 0.26x) dx
The anti-derivative of x(0.76 - 0.26x) is
0.76/2 * x^2 - 0.26/3 * x^3
Evaluated from 0 to 2 this is:
0.76/2 * 2^2 - 0.26/3 * 2^3 = .0841
Answer: .0841
5. E(X^2) = integral (from 0 to 2) x^2(0.76 - 0.26x) dx
The anti-derivative of 0.76x^2 - 0.26x^3 is:
0.76/3 * x^3 - 0.26/4 * x^4
Evaluated from 0 to 2 this is:
0.76/3 * 2^3 - 0.26/4 * 2^4 = .0276
Answer: .0276
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