What does "normalization" mean and how to verify that a sample or a distribution is normalized? I have a question in which it asks to verify whether if the Uniform distribution (${\rm Uniform}(a,b)$) is normalized. 


*

*For one, what does it mean for any distribution to be normalized? 

*And two, how do we go about verifying whether a distribution is normalized or not?  


I understand by computing 
$$
\frac{X-\text{mean}}{\text{sd}}
$$ 
we get normalized data, but here it's asking to verify whether a distribution is normalized or not.
 A: Unfortunately, terms are used differently in different fields, by different people within the same field, etc., so I'm not sure how well this can be answered for you here.  You should make sure you know the definition that your instructor / the textbook is using for "normalized".  However, here are some common definitions:  
Centered:
$$
X-{\rm mean}
$$
Standardized:
$$
\frac{X-\text{mean}}{\text{sd}}
$$
Normalized:
$$
\frac{X-\min(X)}{\max(X)-\min(X)}
$$
Normalizing in this sense rescales your data to the unit interval.  Standardizing turns your data into $z$-scores, as @Jeff notes.  And centering just makes the mean of your data equal to $0$.  
It is worth recognizing here that all three of these are linear transformations; as such, they do not change the shape of your distribution.  That is, sometimes people call the $z$-score transformation "normalizing" and believe, because of $z$-scores' association with the normal distribution, that this has made their data normally distributed.  This is not so (as @Jeff also notes, and as you could tell by plotting your data before and after).  Should you be interested, you could change the shape of your data using the Box-Cox family of transformations, for example.  
With respect to how you could verify these transformations, it depends on what exactly is meant by that.  If they mean simply to check that the code ran properly, you could check means, SDs, minimums, and maximums.  
A: By using the formula you provided on each score in your sample, you are converting them all to  z-scores.
To verify that you computed all the z-scores correctly, find the new mean and standard deviation of your sample. If the mean is $0$ and the standard deviation is $1$, you've done everything correctly.
The purpose of doing this is to put everything in units relative to the standard deviation of your sample. This may be useful for a variety of purposes, such as comparing two different data sets that were scored using different units (centimeters and inches, perhaps).
It is important not to get this confused with asking whether a distribution is normal, i.e. whether it approximates a Gaussian distribution.
A: After consulting the TA, what the question was asking was whether if 
$$ 
\int_{-\infty}^{\infty}f(x)dx=1
$$
where $f(x)$ in this case is the density of the uniform(a,b).
