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I am trying to solve some of my past papers exam and really don't know where I did wrong here :

enter image description here

Naturally, I know that the integral from minus infinity to infinity for any pdf equals $1$. I solved this :

$$\int_1^a\frac{\ln(0.5)}{x}\,\mathrm{d}x=1$$

But when I did, I came with a value of a smaller than $1$ which is not possible... ($a=e^{-1/\ln(2)}=0.27$) I really don't get where I did wrong. Obviously when I tried to find the CDF by integration, I found something weird too...

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    $\begingroup$ note that $log(.5) <0 $ and $\int_1^a = - \int_a^1$ $\endgroup$ – meh Apr 6 '18 at 17:03
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    $\begingroup$ @aginensky But we are told $a \ge 1$! The fact is that since $\log(0.5)\lt 0,$ $f$ cannot possibly be a probability density for positive $x$--but we are also told $x \ge 1.$ $\endgroup$ – whuber Apr 6 '18 at 17:04
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    $\begingroup$ well then $log(.5) <0 $ I'm saying the problem is wrong- final answer. $\endgroup$ – meh Apr 6 '18 at 17:06
  • $\begingroup$ I think you miscalculated. I get approximately 1.275 $\endgroup$ – Michael R. Chernick Apr 6 '18 at 17:27
  • $\begingroup$ I did it by hand first and found 0.237. Wolfram Alpha and my calculator gave me the same answer. $\endgroup$ – Student number x Apr 6 '18 at 17:38
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You should have started with drawing this alleged PDF:

enter image description here

It's always negative in $x\in[1,\infty)$. Just a couple of weeks ago someone was asking here whether the probability is the study of nonnegative functions.

Someone was having hangover when coming up with the problem :)

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