Example normalizing and analyzing continuous probability

In [50]:

    

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad


    

In [51]:

    

def phif(T):
    return np.exp(-(T-20.)**2/50.)


    

In [52]:

    

T = np.linspace(-20,50,1000)
phi = phif(T)
plt.plot(T,phi)
plt.xlabel('Temperature (C)')
plt.ylabel('Relative probability (1/C)')


    

    
Out[52]:





Text(0, 0.5, 'Relative probability (1/C)')






    

In [53]:

    

N,err = quad(phif,-20,50)

def phif_norm(T):
    return phif(T)/N

plt.plot(T,phif_norm(T))
plt.xlabel('Temperature (C)')
plt.ylabel('Relative probability (1/C)')


    

    
Out[53]:





Text(0, 0.5, 'Relative probability (1/C)')






    

In [54]:

    

def Tphif(T):
    return T*phif_norm(T)
Tbar,err = quad(Tphif,-20,50)
print("Expectation value of T = {:4.1f} deg C".format(Tbar))

plt.plot(T,phif_norm(T))
plt.xlabel('Temperature (C)')
plt.ylabel('Relative probability (1/C)')
plt.plot([Tbar,Tbar],[0,0.09],ls='-')


    

    




Expectation value of T = 20.0 deg C






    
Out[54]:





[<matplotlib.lines.Line2D at 0x151e1ef3d0>]






    

In [55]:

    

def T2phif(T):
    return T*T*phif_norm(T)
T2bar,err = quad(T2phif,-20,50)
print("Expectation value of T**2 = {:4.1f} deg C**2".format(T2bar))

Trms=np.sqrt(T2bar)
print("Room mean square T ={:4.1f} deg C".format(Trms))

variance = T2bar-Tbar*Tbar
print("Variance ={:4.1f} deg C**2".format(variance))

stddev = np.sqrt(variance)
print("Standard deviation ={:4.1f} deg C".format(stddev))


    

    




Expectation value of T**2 = 425.0 deg C**2
Room mean square T =20.6 deg C
Variance =25.0 deg C**2
Standard deviation = 5.0 deg C

In [56]:

    

prob,err = quad(phif_norm,Tbar-stddev,Tbar+stddev)
print("Probability = {:4.3f}".format(prob))


    

    




Probability = 0.683

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