问题及解答

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?
   

(a)

Since the pdf $f(x)$ should satisfy the condition $\int_0^2 f(x)dx=1$, we have

\[
\begin{split}
\int_0^2 kx^2dx&=1\Rightarrow k\int_0^2 x^2dx=1\\
&\Rightarrow k\cdot\frac{1}{3}x^3\biggr|_0^2=1\\
&\Rightarrow k\cdot\frac{8}{3}=1\\
&\Rightarrow k=\frac{3}{8}.
\end{split}
\]

(b)

The probability that the lecture ends within $1$ min of the bell ringing is

\[
\begin{split}
P(X\leqslant 1)&=\int_0^1 f(x)dx=\int_0^1 kx^2dx=\int_0^1\frac{3}{8}x^2dx\\
&=\frac{3}{8}\cdot\frac{1}{3}x^3\biggr|_0^1\\
&=\frac{1}{8}=0.125
\end{split}
\]

(c)

The probability that the lecture continues beyond the bell for between 60 and 90 sec is

\[
\begin{split}
P(1\leqslant X\leqslant\frac{3}{2})&=\int_{1}^{\frac{3}{2}}f(x)dx=\int_{1}^{\frac{3}{2}}kx^2dx=\int_{1}^{\frac{3}{2}}\frac{3}{8}x^2dx\\
&=\frac{3}{8}\cdot\frac{1}{3}x^3\biggr|_{1}^{\frac{3}{2}}\\
&=\frac{1}{8}\cdot\Bigl((\frac{3}{2})^3-1^3\Bigr)\\
&=\frac{19}{64}=0.296875
\end{split}
\]

(d)

The probability that the lecture continues for at least 90 sec beyond the bell is

\[
\begin{split}
P(X\geqslant\frac{3}{2})&=\int_{\frac{3}{2}}^{2}f(x)dx=\int_{\frac{3}{2}}^{2}kx^2dx=\int_{\frac{3}{2}}^{2}\frac{3}{8}x^2dx\\
&=\frac{3}{8}\cdot\frac{1}{3}x^3\biggr|_{\frac{3}{2}}^{2}\\
&=\frac{1}{8}\Bigl(2^3-(\frac{3}{2})^3\Bigr)\\
&=\frac{37}{64}=0.578125
\end{split}
\]