What conditions must be satisfied by parameters $a,b\in \mathbb{R}$ if function $F:\mathbb{R} \rightarrow \mathbb{R}$ given by an equation: $$F(t)=\begin{cases} 0 \ for \ t<1 \\ a+\frac{b}{t} \ for \ t\ge1 \end{cases}$$ is cumulative distribution function? What should be assumed to make this distribution continuous? I know the definition of a cumulative distribution function, but i have no idea how to apply them in this case.

  • 2
    $\begingroup$ I think just $a=1$ and $-1\leq b\leq0$ will do $\endgroup$ – MPW Nov 5 '15 at 17:54
  • $\begingroup$ $F$ is continuous on $(-\infty,1)$ and on $(1,\infty)$. So the only thing needed for making it continuous on $\mathbb R$ is continuity at $1$. $\endgroup$ – drhab Nov 5 '15 at 18:10
  • $\begingroup$ I am not certain whether i fully understand why $-1\leq b\leq0$? $\endgroup$ – mkropkowski Nov 5 '15 at 18:31

$F(t)$ must tend to $1$ as $t\to\infty$. This forces $a=1$. Also, $b$ cannot be positive, or else $F$ becomes a decreasing function. Finally, the value of $F(t)$ must be nonnegative for all $t$, which forces $b\ge -1$ (plug $t=1$).


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