def _pln_pdf(x, alpha, nu, tau2):
    A1 = np.exp(alpha * nu + alpha ** 2 * tau2 / 2)
    fofx = alpha * A1 * x ** (-alpha - 1) *\
        norm.cdf((np.log(x) - nu - alpha * tau2) / np.sqrt(tau2))
    return fofx