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) = 1 - 0.5x
0 < x < 2

a) What is the probability a trainee will complete the task in less than 1.19 minutes? Give your answer to four decimal places.

b) What is the probability that a trainee will complete the task in more than 1.19 minutes? Give your answer to four decimal places.

c) What is the probability it will take a trainee between 0.27 minutes and 1.19 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

Answer 1

a)

P(X<1.19)= f(x) dx = (1-0.5x) dx =(x-0.5x^2/2) |^1.19₀ =0.8360

b)

P(X>1.9)=1-0.8360=0.1640

c)

P(0.27<X<1.19)=0.5842

d)

E(X)= xf(x) dx = (x-0.5x^2) dx =(x^2/2-0.5x^3/3) |²₀ =2/3 =0.6667

e)

E(X²)= x^2f(x) dx = (x^2-0.5x^3) dx =(x^3/3-0.5x^4/4) |²₀ =2/3 =0.6667

f)

Var(X)=E(X²)-(E(X))² =0.2222