Your answer is correct; the provided answer is the MLE of $\theta + 1$, not $\theta$.  You can see this either through simulation, or by direct calculation:  if we let $$s = -\sum_{i=1}^n \log y_i &gt; 0,$$ then $$\hat \theta = -1 + \frac{n}{s},$$ and $$\ell(\hat\theta \mid s) = n \log(\hat \theta + 1) - \hat\theta s = s + n \log \tfrac{n}{s} - n.$$  But $$\ell(\hat \theta + 1 \mid s) = n \log \left(\tfrac{n}{s} + 1\right) - n$$  So their difference is $$d(s,n) = \ell(\hat\theta \mid s) - \ell(\hat\theta + 1 \mid s) = s + n \log \frac{n}{n+s}.$$  It is easy to show that this expression is always positive for $s &gt; 0$ and $n = 1, 2, 3, \ldots$:  its derivative with respect to $s$ is $\frac{s}{n+s} &gt; 0$, so it is strictly increasing, and $d(0,n) = 0$.  This proves that the estimator $n/s$ yields a strictly smaller log-likelihood than your correct MLE $\hat \theta = -1 + n/s$.

Finding the MLE for $θ$ given a probability density function $f(y |\theta )$.
When I searched for the social engineering toolkit I got the error "no such path or file".

I used this orders for installing it but without success.


  be sure to install subversion before installing the selected tools.


(i did it)

svn co http://svn.secmaniac.com/social_engineering_toolkit set/


and the link is not working anymore.
