Chapter 5.9, Problem 60E

To determine

(a)

The inverse function $t(F)$

To determine

(b)

To justify the approximation $M\approx \frac{1}{N}\sum _{j=1}^{N}t(\frac{j}{N})$. Then, argue, by passing to the limit as $N\to \infty$that $M={\displaystyle \underset{0}{\overset{1}{\int }}t(F)dF}$. Strictly speaking, this is an improper integral because $t(0)$ is infinite. Hence, we define $M$ as a limit

$M=\underset{c\to 0}{\overset{}{\lim }}{\displaystyle \underset{c}{\overset{1}{\int }}t(F)dF}$

To determine

(c)

To verify the formula $M={\displaystyle \underset{}{\overset{}{\int }}\ln xdx}=x\ln x-x$by differentiation and use it to show that for $c>0$

$M=\underset{c\to \infty }{\lim }(\frac{1}{k}+\frac{1}{k}(c\ln c-c))$

To determine

(d)

To show that $M=1/k$ by evaluating the limit (use L’Hospital’s Rule to compute $\underset{c\to 0}{\lim }c\ln c$)

To determine

(e)

The mean time to decay for radon (with a half-life of 3.825 days).