Since \widehat{\beta} = \widehat{\beta}(\boldsymbol{X}) is a function of random variables, \widehat{\beta} itself must be a random variable. Thus it has a pdf, call this function p(\boldsymbol{t}) . The aim of the bootstrap is to estimate p(\boldsymbol{t}) by the relative frequency of \widehat{\beta} . You can think of this as using a histogram in the place of p(\boldsymbol{t}) . If the relative frequency closely resembles p(\vec{t}) , then using numerics, it is straight forward to estimate all the interesting parameters of p(\boldsymbol{t}) using point estimators.