# Regarding custom wavelets and their validity

I've recently started working with wavelets, and want to focus primarily on Discrete Wavelet Transform. For experimental purposes, I use the PyWavelets library in Python. In one of my tests, I tried the following code.

Basically, I randomly sample the coefficients for the lowpass and highpass filters from a normal distribution, and use those coefficients to plot the scaling and wavelet functions.

import matplotlib.pyplot as plt
import pywt
import numpy as np

fig = plt.figure()
fig.subplots_adjust(hspace=0.2, wspace=0.2, bottom=.02, left=.06,
right=.97, top=.94)

dec_lo, dec_hi, rec_lo, rec_hi = np.random.normal(0.0, 0.5, (4, 2)).tolist()
print(dec_lo, ' ', dec_hi, ' ', rec_lo, ' ', rec_hi)
filter_bank = [dec_lo, dec_hi, rec_lo, rec_hi]
wavelet = pywt.Wavelet(name="myWavelet", filter_bank=filter_bank)

phi, psi, _, _, x = wavelet.wavefun(level=5)

color = 'red'
ax = fig.add_subplot(1, 2, 1)
ax.set_title(wavelet.name + " phi")
ax.plot(x, phi, color)
ax.set_xlim(min(x), max(x))

ax = fig.add_subplot(1, 2, 2)
ax.set_title(wavelet.name + " psi")
ax.plot(x, psi, color)
ax.set_xlim(min(x), max(x))

plt.show()


A run of the above code gave me the following plot (Since values are randomly sampled, different runs will give different plots):

My question is, can it be shown that these plots are valid scaling and wavelet functions? Or are they simply meaningless plots generated for the sampled coefficients?