Trouble finding CDF from PDF

Question

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

@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 $x \ge 1.$ — whuber, Apr 06 '18 at 17:04
well then $log(.5) <0 $ I'm saying the problem is wrong- final answer. — meh, Apr 06 '18 at 17:06
I did it by hand first and found 0.237. Wolfram Alpha and my calculator gave me the same answer. — Student number x, Apr 06 '18 at 17:38

Aksakal · Accepted Answer · 2018-04-06T19:03:37.730

5

You should have started with drawing this alleged PDF:

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 :)

edited Apr 06 '18 at 19:03

answered Apr 06 '18 at 18:57

Aksakal

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Trouble finding CDF from PDF

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