Call a non-negative integer less than or equal to $10000$ for which the digit $1$ does not appear in its decimal expansion a good integer.
Let us denote by $A$ the average value of all good integers. Since $10000$ is not a good integer, it suffices to consider only good integers of $4$ or less digits. Write a good integer in the form $a_1 + 10a_2 + 10^2a_3 + 10^3a_4$, where $a_j$'s are chosen from the set $\{0, 2, 3, 4, 5, 6, 7, 8, 9\}$.
Since for $a_1$ each of the values $0, 2, 3, \ldots, 9$ can be chosen the same number of times, the average value for $a_1$ over the set of all good integers is $\frac{1}{9}(0 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = \frac{44}{9}$.
Similarly, the average value of $a_2, a_3, a_4$ over the set of all good integers is $\frac{44}{9}$ for each one. Hence we get
$$A = \frac{44}{9}(1 + 10 + 10^2 + 10^3) = \frac{48884}{9}.$$