The time, in 100 hours, that a student uses her game console over a year is a random variable X with probability density function

f(x)=x if 0 f(x)=2−x if 1≤x<2
f(x)=0 otherwise.

The power (in number of kilowatt hours) expended by the student's game console each year is 50X2+28. For these problems, please ensure your answers are accurate to within 3 decimals.

Find the mean amount of power expended by the student's game console per year.

The mean amount of power expended by the student's game console per year = 166.889 kWh

Step-by-step explanation:

The mean amount of time that the student uses her game is first obtained, using expected values calculation.

E(X) = Σ xᵢpᵢ = ∫

xᵢ = each variable

pᵢ = probability of each variable

In terms of integrals, the sum can be made over the whole possible sample space, given that pᵢ = f(x)

E(X) = Σ xᵢpᵢ = ∫ x f(x) dx with integral done over the whole interval available.

f(x)=x if 0 < x < 1

f(x)=2−x if 1≤x<2

f(x)=0 otherwise

E(X) = ∫ x f(x) dx = ∫¹₀ x (x) dx + ∫²₀ x (2 - x) dx

= ∫¹₀ x² dx + ∫²₀ (2x - x²) dx

= [x³/3]¹₀ + [x² - (x³/3)]²₀

= (1/3) + [4 - (8/3)] = (1/3) + (4/3) = (5/3)

Expected number of hours that the student uses her game = (5/3) hundred hours. = 166.67 hours

The power in kilowatts-hour consumed by the console is given by

50X² + 28

E(X) = (5/3) hundred hours

Power = 50[E(X)]² + 28 = 50[(5/3)]² + 28

Power consumed = 166.889 kWh

Hope this Helps!!!