Any positive integer can be expressed as k*2^n, where k is an odd number and n is a natural number. Each operation of the Collatz conjecture can be represented by (3k+1)*2^n, regardless of whether it is an odd or even number. The distribution type of k belongs to deterministic random distribution. Let 2^t be a perfect square number that is just less than 3k, and the cumulative probability value of (3k+1) being a perfect square number after each operation in the Collatz conjecture is conservatively estimated as Σ1/2^t. By comparing with the harmonic function Σ1/n, it is proved that when the number of operations gradually increases, the cumulative probability function value Σ1/2^t of (3k+1) being a perfect square number is much larger than 1, and tends to infinity when the number of operations is infinitely large. This result shows that the occurrence of (3k+1) being a perfect square is inevitable, thus proving the Collatz conjecture.