Chapter 18, Problem 45RE

(a)

To determine

To show: The probability $P\left(0<X\le b\right)$ can be rewritten as $\frac{1}{\sqrt{2\pi }}\left[{2x^{\frac{1}{2}}{e}^{-\frac{x}{2}}|}_0^b+{\displaystyle \underset{0}{\overset{b}{\int }}{x}^{\frac{1}{2}}{e}^{-\frac{x}{2}}dx}\right]$

(b)

To determine

To approximate: The value of the probability of $P\left(0<X\le 1\right)$ using Simpson’s rule with $n=12$.

(c)

To determine

To approximate: The value of the probability of $P\left(0<X\le 10\right)$ using Simpson’s rule with $n=12$.

(d)

To determine

To find: The value of $\frac{1}{\sqrt{2\pi }}\left[{2x^{\frac{1}{2}}{e}^{-\frac{x}{2}}|}_0^b+{\displaystyle \underset{0}{\overset{b}{\int }}{x}^{\frac{1}{2}}{e}^{-\frac{x}{2}}dx}\right]$ as $b\to \infty$ and check whether the results from parts (b) and (c) supports or not.