Math Problems Vault

Problem 4.1.1

(a)(b)(c)

For what values of the parameters are the following functions probability density functions?

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(a)

$f(x) = C\left\{ x(1-x)\right\}^{-\frac{1}{2}}, 0 < x < 1$ , the density function of the 'arc sine law'.

(b)

$f(x) = C \exp \left(-x-e^{-x}\right), x \in \mathbb {R}$ , the density function of the 'extreme-value distribution'.

(c)

$f(x) = C\left(1+x^{2}\right)^{-m}, x \in \mathbb {R}$ .

Problem 4.1.2

Find the density function of $Y = a X$ , where $a > 0$ , in terms of the density function of $X$ . Show that the continuous random variables $X$ and $-X$ have the same distribution function if and only if $f_{X}(x) = f_{X}(-x)$ for all $x \in \mathbb {R}$ .

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Problem 4.1.3

If $f$ and $g$ are density functions of random variables $X$ and $Y$ , show that $\alpha f+(1-\alpha ) g$ is a density function for $0 \leq \alpha \leq 1$ , and describe a random variable of which it is the density function.

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Problem 4.1.4

(a)(b)(c)(d)

Let $X$ be a positive random variable with density function $f$ and distribution function $F$ . Define the hazard function $H(x) = -\log [1-F(x)]$ and the hazard rate

r(x) = \lim _{h \downarrow 0} \frac{1}{h} \mathbb {P}\left(X \leq x+h \mid X > x\right), \quad x \geq 0

Show that:

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(a)

$r(x) = H^{\prime }(x) = f(x) /\left(1-F(x)\right)$ ,

(b)

If $r(x)$ increases with $x$ then $H(x) / x$ increases with $x$ ,

(c)

$H(x) / x$ increases with $x$ if and only if $[1-F(x)]^{\alpha } \leq 1-F(\alpha x)$ for all $0 \leq \alpha \leq 1$ ,

(d)

If $H(x) / x$ increases with $x$ , then $H(x+y) \geq H(x)+H(y)$ for all $x, y \geq 0$ .