How to integrate a function defined by an array

Question

I constructed and plotted a function defined by an array, according to the following code:

# set parameters
mean   = np.array([0, 3])   
sigma  = np.array([1, .8])  
weight = np.array([.4, .6])  
factor = 4
points = 1000

# set limits for plot
min_plot = (mean - (factor*sigma)).min()
max_plot = (mean + (factor*sigma)).max()

# set normal  distributions
d0 = stats.norm(mean[0], sigma[0])
d1 = stats.norm(mean[1], sigma[1])

# set mixed normal
data = np.array([d0.pdf(x), d1.pdf(x)])
y = np.dot(weight, data)

# displays
x = np.linspace(min_plot, max_plot, points)
plt.plot(x, y, '-', color = 'black', label='Normal mixed')

Which gave me the following plot:

Please, what would be the simplest way to integrate $y$ between two given values, for example $x=2$ and $x=4$? I am aware of scipy.integrate, but don't understand how to use it in this particular case...

Answer 1

You need to define a function. Then you can use some numerical method for integration of that function within the specified boundaries.

I am giving an example with scipy.integrate.quad:

from scipy.integrate import quad
from scipy import stats

# define function to be integrated:
def f(x):
    
    mean   = np.array([0, 3])   
    sigma  = np.array([1, .8])  
    weight = np.array([.4, .6])  
    factor = 4
    points = 1000
    
    # set normal  distributions
    d0 = stats.norm(mean[0], sigma[0])
    d1 = stats.norm(mean[1], sigma[1])

    # set mixed normal
    data = np.array([d0.pdf(x), d1.pdf(x)])
    y = np.dot(weight, data)
    
    return y

# integrate function from 2 to 4
quad(f, 2, 4)

returns (0.4823076558823121, 5.354690645135298e-15), i.e. the integral and the absolute error associated with the interval.