  In this paper, we study some important statistics of the random graph in the
directed preferential attachment model introduced by B. Bollob\'as, C. Borgs,
J. Chayes and O. Riordan. First, we find a new asymptotic formula for the
expectation of the number $ n_{in}(d,t) $ of nodes of a given in-degree $ d $
in a graph in this model with $t$ edges, which covers all possible degrees. The
out-degree distribution in the model is symmetrical to the in-degree
distribution. Then we prove tight concentration for $n_{in}(d,t)$ while $d$
grows up to the moment when $n_{in}(d,t)$ decreases to $\ln^2 t$; if $d$ grows
even faster, $n_{in}(d,t)$ is zero \textbf{whp}. Furthermore, we study a more
complicated statistic of the graph: $ X(d_1,d_2,t) $ is the total number of
edges from a vertex of out-degree $d_1$ to a vertex of in-degree $d_2$. We also
find an asymptotic formula for the expectation of $ X(d_1,d_2,t) $ and prove a
tight concentration result.
