A College Professor Always Finishes His Lectures Within 2 Minutes $f ( x ) = \left\{ \begin{array} { l l } k x ^ { 2 } & 0 \leq x \leq 2 \\0 & \text { otherwise }\end{array} \right.$

Essay

A college professor always finishes his lectures within 2 minutes after the bell rings to end the period and the end of the lecture. Let X = the time that elapses between the bell and the end of the lecture and suppose the pdf of X is $f ( x ) = \left\{ \begin{array} { l l } k x ^ { 2 } & 0 \leq x \leq 2 \\0 & \text { otherwise }\end{array} \right.$
a. Find the value of k. [Hint: Total area under the graph of f(x) is 1.]
b. What is the probability that the lecture ends within 1minutes of the bell ringing?
c. What is the probability that the lecture continues beyond the bell for between 60 and 90 seconds?
d. What is the probability that the lecture continues for at least 90 seconds beyond the bell?