[Exer9-3] Exercise 5 of Book {Devore2017B} P.148
A college professor never finishes his lecture before the bell rings to end the period and always finishes his lectures within $2$ min after the bell rings. Let $X$= the time that elapses between the bell and the end of the lecture and suppose the pdf of $X$ is
\[
f(x)=
\begin{cases}
kx^2, & 0\leqslant x\leqslant 2, \\
0, & \mbox{otherwise}.
\end{cases}
\]
- (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 $1$ min of the bell ringing?
- (c) What is the probability that the lecture continues beyond the bell for between $60$ and $90$ sec?
- (d) What is the probability that the lecture continues for at least $90$ sec beyond the bell?