How to show a random variable is power law distributed and estimate its parameter? According to this the power-law distribution can be written as $$P(X=f) = \frac{1}{Z}f^{1+\frac{1}{b}}$$ where $f$ is the frequency of random variable $X$, $Z$ is the normalizing constant, and $b$ is the parameter of the power-law distribution. How can I estimate the $b$ parameter?
 A: The answer is rather complicated. This paper, Clauset et al, "Power law distributions in empirical data," walks through a number of methods and provides good commentary on why some approaches are better than others.
The paramaterization that you provide is not one that I typically encounter, though it should be clear how it relates to the estimator below. More usually, the power law probability distribution has form
$$p(x) \propto x^{-\alpha}$$
I would recommend focusing on this form, because is is used in the Clauset article, which is the most complete treatment of your question that I am aware of.
The MLE of the power law distribution for continuous data is
$$
\hat{\alpha}=1+n\left[\sum_{i=1}^n\ln\frac{x_i}{x_{min}}\right]^{-1}
$$
The MLE in the discrete case is somewhat more involved, but an approximation to the exact MLE has a form very similar to that of the continuous case.
While plotting data that you suspect to be power law on log axes will probably resemble a straight line. Some authors just that as a justification for using a linear regression to estimate the scaling parameters. Clauset et al recommend against that procedure, as its estimates are not nearly as good as the MLE or other methods. There is a discussion of the disadvantages in the paper.
Considering whether the power law fit to the data is plausible is covered in section 4 of the paper. The authors suggest a Monte Carlo procedure to estimating the p-value.
Superficially, it might seem helpful to use something like a Kolmogorov-Smirnov test of the data sample against the model attained by parameters directly estimated from the data. This is an incorrect application of the test, though, because the test assumes the model is fully specified, i.e. not estimated from the data. In the case where the data is used to estimate the parameters, there will be a natural and obvious dependence between the data and those parameters, so the test will be biased.
