A new scheme for the polynomial chaos expansion (PCE) is developed and tested to quantify the propagation of the input uncertainty in CFD. The Haar wavelet is used as basis for the PCE instead of globally continuous, orthogonal polynomials to properly represent the strong discontinuity such as a shock wave. We further extended the method to inputs with arbitrary probability distribution functions (PDF), although the natural application of the Haar wavelet is the uniform PDF. A transonic nozzle flow with an uncertainty in the nozzle-exit pressure is simulated. The wavelet basis well reproduces the Monte Carlo simulations at a single execution of the program, unlike the multi-element (ME) PCE developed in the previous report. But the quantitative agreements of the statistics are better obtained by the result using ME-PCE.