The uniform distribution models a random variable, X, that can take any value within the range [a, b] with equal probability.

The PDF is given by  for a ≤ x ≤ b, and 0 otherwise.

The expectation and variance are given respectively by the following expressions:

A continuous uniform probability distribution is generated and plotted for various sample sizes in the following code:

    In [11]: np.random.seed(100)  # seed the random number generator
                                  # so plots are reproducible
             subplots = [111,211,311]
             ctr = 0 
             fig, ax = plt.subplots(len(subplots), figsize=(10,12))
             nsteps=10
             for i in range(0,3):
               cud = np.random.uniform(0,1,nsteps) # generate distrib
               count, bins, ignored = ax[ctr].hist(cud,15,normed=True)
               ax[ctr].plot(bins,np.ones_like(bins),linewidth=2, color='r')
               ax[ctr].set_title('sample size=%s' % nsteps)
               ctr += 1
               nsteps *= 100
             fig.subplots_adjust(hspace=0.4)
             plt.suptitle("Continuous Uniform probability distributions for various sample sizes" , fontsize=14)
  

The following is the output: