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) = 0.61 - 0.11x
0 < x < 2

a) What is the probability a trainee will complete the task in less than 1.43 minutes? Give your answer to four decimal places.  

b) What is the probability that a trainee will complete the task in more than 1.43 minutes? Give your answer to four decimal places.  

c) What is the probability it will take a trainee between 0.47 minutes and 1.43 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<1.43)= (0.61-0.11x) dx =0.61x-0.11x²/2 |^1.43₀ =0.7598

b)

probability that a trainee will complete the task in more than 1.43 minutes =P(X>1.43)=1-0.7598 =0.2402

c)

probability it will take a trainee between 0.47 minutes and 1.43 minutes =P(0.47 <X<1.43)=0.61x-0.11x²/2 |^1.43_0.47 =0.4853

d)

expected time E(X)= xf(x) dx = (0.61x-0.11x²) dx =0.61x²/2-0.11x³/3 |²₀ =0.9267

e)

E(X²) = x²f(x) dx = (0.61x²-0.11x³) dx =0.61x³/3-0.11x⁴/4 |²₀ =1.1867

f)Var(X)=E(X²)-(E(X))² =0.3280 ( please try 0.3279 if this comes wrong)