Section 2.4

Worked examples

[2/100%]

Show

LaTeX

Show LaTeX

Problem 2.4.1

Let $X$ be a random variable with a continuous distribution function $F$ . Find expressions for the distribution functions of the following random variables:

(a)

$X^{2}$ ,

(b)

$\sqrt{X}$ ,

(c)

$\sin X$ ,

(d)

$G^{-1}(X)$ ,

(e)

$F(X)$ ,

(f)

$G^{-1}(F(X))$ , where $G$ is a continuous and strictly increasing function.

Problem 2.4.2

Let $X$ be a random variable with distribution function $F$ , and let $a < b$ . Sketch the distribution functions of the 'truncated' random variables $Y$ and $Z$ given by

Y = \begin{cases} a & \text{ if } X < a, \\ X & \text{ if } a \leq X \leq b, \\ b & \text{ if } X > b, \end{cases} \quad Z = \begin{cases} X & \text{ if }\left|X\right| \leq b \\ 0 & \text{ if }\left|X\right| > b\end{cases}

Indicate how these distribution functions behave as $a \rightarrow -\infty , b \rightarrow \infty$ .