Question

1. A college professor never finishes his lectures before the end of the hour but always finishes his lectures within 5 minutes after the hour. Let X = the amount of over-time the professor lectures for, with pdf (probability density function) kx2 0 < x < 5 otherwise a. Find the value of k and draw the corresponding density curve between 0 and 5. (Remember that the total area under the graph of f(x) is 1.) b. Which of the following is true? In general, not just for this pdf (probability density function), the highest f(x) the curve achieves must be exactly 1. The highest f(x) the curve achieves must be $ 1$. The highest f(x) the curve achieves must be > 1. c. What is the probability that the lecture ends within 1 minute of the end of the hour? d. What is the probability that the lecture continues beyond the hour for between 60 and 90 seconds? e. What is the probability that the lecture continues for at least 90 seconds beyond the end of the hour?

Answer 1

a) for this to be valid:

5
f(x )dx =
5
kx² dx =k*(x³/3) |⁵₀ =k*125/3 =1

k=3/125

b)

the highest f(x) of the curve achieves must be 1

c)

P(X<1) =$\int_{0}^{1}$ f(x )dx =$\int_{0}^{1}$ (3/125)x² dx =(1/125)*(x³) |¹₀ =1/125

d)

P(1 <X<1.5 )=(1/125)*(x³) |^1.5₁ =0.027-0.008 =0.0190

e)

P(X>1.5)=1-P(X<1.5) =1-(1/125)*(x³) |^1.5₀ =1-0.027 =0.9730