Chapter 10: Q. 10.2 (page 430)

Develop a technique for simulating a random variable having density function
$f (x) \{\begin{cases} e^{2 x} & - \infty < x < 0 \\ e^{- 2 x} & 0 < x < \infty \end{cases}$

Short Answer

Expert verified

The information for the function will get by using the universality of uniform distribution.

Step by step solution

Given Information

We have given a function

$f (x) \{\begin{cases} e^{2 x} & - \infty < x < 0 \\ e^{- 2 x} & 0 < x < \infty \end{cases}$ .

Simplify

We have

$F (x) = \int_{- \infty}^{x} e^{2_{s}} d s = \frac{e^{2 x}}{2}$

and for $x > 0$ we have

$F (x) = \int_{- \infty}^{0} e^{2_{s}} d s + \int_{0}^{x} e^{- 2_{s}} d s = 1 - \frac{e^{- 2 x}}{2}$

Calculating the value $F^{- 1}$ . For $Y \in (0, 1 / 2)$ we have

$Y = \frac{e^{2 x}}{2} \Rightarrow x = \frac{1}{2} \log (2 Y)$

and for $Y \in (1 / 2, 1)$ we have that

$Y = 1 - \frac{e^{- 2 x}}{2} \Rightarrow x = - \frac{1}{2} \log (2 - 2 Y)$

Choose random number (call it $Y$ ) from the interval $(0, 1)$ . if it is less than $\frac{1}{2}$ , declare $x = \frac{1}{2} \log (2 Y)$ .If it is greater than $\frac{1}{2}$ , declare $x = - \frac{1}{2} \log (2 - 2 Y)$ . From the universality of the Uniform, we have $x$ follows the distribution of $X$ .

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Develop a technique for simulating a random variable having density function
$f (x) \{\begin{cases} e^{2 x} & - \infty < x < 0 \\ e^{- 2 x} & 0 < x < \infty \end{cases}$

Short Answer

Step by step solution

Given Information

Simplify

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Step by step solution

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Simplify

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Develop a technique for simulating a random variable having density function
$f (x) \{\begin{cases} e^{2 x} & - \infty < x < 0 \\ e^{- 2 x} & 0 < x < \infty \end{cases}$