  An elementary recursive relation for M$\ddot{\mathrm{o}}$bius function $\mu
(n)$ is introduced by two simple ways. With this recursive relation, $\mu (n)$
can be calculated without directly knowing the factorization of the $n$. $\mu
(1) \sim \mu (2 \times 10^7) $ are calculated recursively one by one. Based on
these $2\times 10^7$ samples, the empirical probabilities of $\mu (n)$ of
taking $-1$, 0, and 1 in classic statistics are calculated and compared with
the theoretical probabilities in number theory. The numerical consistency
between these two kinds of probability show that $\mu (n)$ could be seen as an
independent random sequence when $n$ is large. The expectation and variance of
the $\mu (n)$ are $0$ and $6 n/ \pi^2$, respectively. Furthermore, we show that
any conjecture of the Mertens type is false in probability sense, and present
an upper bound for cumulative sums of $\mu (n)$ with a certain probability.
