This paper discusses geometric programs with joint probabilistic constraints. When the stochastic parameters are normally distributed and independent of each other, we approximate the problem by using piecewise polynomial functions with non-negative coefficients, and transform the approximation problem into a convex geometric program. We prove that this approximation method provides a lower bound. Then, we use an improved Bonferroni approximation method to find an upper bound. Finally, numerical tests are carried out with a shape optimization problem.