A wide variety of probabilistic models can be encapsulated in the framework of a Gaussian vector stochastically scaled by a positive random variable. Although conceptually straightforward, in all but a few cases the resultant random vectors posses density functions which are either computationally intractable or completely unknown. In this work we consider the problem of density estimation for these random vectors. We first show how the problem can be reduced to sampling from the positive real numbers and evaluating the multivariate Gaussian density, yielding a standard Monte Carlo estimator. We also demonstrate how significantly higher accuracy can be achieved, at roughly the same computational cost, by employing the theory of random Riemann sums. Both of these approaches yield unbiased estimates of the likelihood; therefore, they enable exact inference despite an intractable, or even unknown, likelihood function by appealing to the recently developed theory of pseudo-marginal Markov chain Monte Carlo.