Radioactive decay is a statistical process i. e we cannot precisely predict the timing of a particular radioactivity of a particular nucleus can disintegrate immediately of it may take infinite time. Simply the probability of the number of nuclei being distegrated at any istant. Can be predicated. The rate at which a particular decay process occurs in a radioactive sample is directly proportional to the number of radioactive nuclei present and thus obeys first order kinetics. The factor $-\frac{dN}{N}$ expresses  the fraction of nuclei decayed in time $dt.t_{\frac{1}{2}}$  is the time in which half of the atoms are decayed and average life is the time for the time for the nucleus to survive before decay.