The more degrees of freedom for a student-t distribution the flatter? If yes, why? Assume we have a student-t distribution. Is the following statement true:
"The more degrees of freedom we have the flatter the curve" ?
If yes/no why?
 A: The meaning of the term "flatter curve" is vague, and may mislead.
From a statistical point of view, we could understand it to mean that excess kurtosis is reduced. The Student's t-distribution has positive excess kurtosis. As its degrees of freedom increase, its peak increases while its tails get lighter. The combined effect, as measured by Pearson's kurtosis coefficient, is to reduce kurtosis. 
But from a geometric point of view, a higher peak and lighter tails may provide to some (me included), an image of a steeper curve, not of a flatter one:

To my eyes, the flatter curve is the $d=1$ not the $d=10$ one. Lowering the degrees of freedom we "push" its peak downwards, while "lifting" its tails. The overall "feeling" is one of a shape more flat then before, in the sense that its extremes have gotten closer.
A: To understand t distribution you can refine its definition.
Suppose you have a i.i.d sample of values. Following the central limit theorem, you know that the distribution of the mean follows a normal distribution. When standardized, it follows a standard normal.
$$\sqrt{n}\frac{\bar{X}-\mu}{\sigma} \sim N(0,1) $$
You estimate $\mu$ with sample mean and the $\sigma^2$ with sample variance. Thus everything should be fine. But it might not be. The reason could be that you have a small sample. 
If you do not know the variance parameter, and you have to estimate it from a small sample you might be way off, since the CLT works on large samples. 
T-distribution incorporates uncertainty regarding the estimation of variance parameter from a small sample. So, the distribution of your statistic is similar to a normal one, but it has fatter tails if your sample is small. It has fat tails because in this way you say 'I am less sure that the statistic is close to the real parameter value'. Having fatter tails, it will also have a smaller maximum. 
Note however that for large samples the t distribution goes toward a normal one, so in practice the normal approximation is safer.
A: "As the degrees of freedom increases, the area in the tails of the t-distribution decreases while the area near the center increases. (The tails consist of the extreme values of the distribution, both negative and positive.)"
Source: http://www.dummies.com/education/math/business-statistics/how-the-number-of-degrees-of-freedom-affect-the-graph-of-a-t-distribution/
