What is the probability distribution of linear formula? What is the distribution name when probability values have a linear increasing of the form:
$p(i)= \frac{2}{N(N+1)}i;  0 \leq i \leq N $
 A: It is a valid distribution if $i$ is integer, otherwise it isn't as @Alexis points out. When integer, if we sum over all possibilities, we get $1$:
$$\sum_{i=1}^N{\frac{2}{N(N+1)}}i=\frac{2}{N(N+1)}\sum_{i=1}^N{}i=\frac{2}{N(N+1)}\frac{N(N+1)}{2}=1$$
Wikipedia entry for List of Probability Distributions doesn't associate this with a special name, however it is being used in some sources as Discrete Triangular Distribution, although the name has also been used to describe the probability distribution of two dice, in which the distribution is symmetric around a mean. As continuous case refers generally to asymmetric version of the triangle shape (which also includes $b=c$, i.e. right triangle), the discrete version can also do this, which makes the name valid practically.
A: I think you intend this to be a discrete distribution.
If $N=5$ you have $P(X = i) = \frac{2i}{30} = \frac{i}{15}.$ Notice that, whatever $N$ you choose, $P(X = 0) = 0.$
You need to check that the probabilities add to $1:$ that is, $\sum_{i=0}^{5} P(X = i) = 1.$
For that, you might want to look at this Wikipedia page.
Here is a plot of the PDF (or PMF) of this distribution.

