问题

Let $X$ denote the amount of space occupied by an article placed in a $1$-$\mathrm{ft}^3$ packing container. The pdf of $X$ is
\[
f(x)=
\begin{cases}
  90x^8(1-x), & 0 < x < 1, \\
  0, & \mbox{otherwise}.
\end{cases}
\]

• (a) Graph the pdf. Then obtain the cdf of $X$ and graph it.
• (b) What is $P(X\leqslant .5)$ [i.e., $F(.5)$]?
• (c) Using part (a), what is $P(.25 < X \leqslant .5)$? What is $P(.25\leqslant X\leqslant .5)$?
• (d) What is the $75$th percentile of the distribution?
• (e) Compute $E(X)$ and $\sigma_X$.
• (f) What is the probability that $X$ is within $1$ standard deviation of its mean value?