import numpy as np import matplotlib.pyplot as plt from matplotlib import style style.use('bmh') #Not to show warning messages (to keep the notebook clean) import warnings warnings.filterwarnings('ignore')

#Example 1 t = np.linspace(0, 20, 2000) # t is defined from 0 to 20 with 2000 points plt.figure(figsize=(16,6)) plt.plot(t, np.sin(t)-2) plt.plot(t, np.cos(t)) plt.xlabel('$t$') plt.ylabel('$x(t)$') plt.title('Plot functions') plt.xlim([0, 20]) plt.grid(True) plt.show()

#Example 2 t = np.linspace(-5, 5, 100) # t is defined from -5 to 5 with 100 points plt.figure(figsize=(16,6)) plt.plot(t, t**2,label='$t^2$') #plt.plot(t, 1/3 * t**3, label='$t^3/3$') plt.xlabel('$t$') plt.ylabel('$x(t)$') plt.grid(True) plt.legend() plt.show()

#Example 2 t = np.linspace(0, 5, 100) # t is defined from -5 to 5 with 100 points plt.figure(figsize=(16,6)) plt.plot(t, np.exp(-t)) #plt.plot(t, 1/3 * t**3, label='$t^3/3$') plt.xlabel('$t$') plt.ylabel('$x(t)$') plt.grid(True) plt.show()

def flip(p): return 'H' if np.random.random() < p else 'T'

### Experiment 1 p=0.65 flip(p)

### Experiment 2 - 10 trials a = [None] * 10 #create an empty list with 10 elements for i in range(10): a[i]=flip(p) print(a)

### Experiment 3 - Empirical probability based on frequency #Number o realizations x_max = 10000 count = 0 #Experiment for x in range(0, x_max): a = flip(p) #print a if a == 'H': count = count + 1 print('Number of heads:', count) print('Number of tails:', x_max - count)

#Probability from experiments: Number of Heads or Tails diveded by the number of realizations print('Empirical frequency') print('Probability (heads):', count / x_max) print('Probability (tails):', (x_max-count) / x_max) print('Bias probability') print('Probability (heads):', p) print('Probability (tails):', 1-p)

#Simulating Poisson distribution with mean 4 mean=4 samples = np.random.poisson(mean, 10) print(samples)

### Drawing a histogram (empirical distribution) mean=2 samples_mean2 = np.random.poisson(mean, 10000) n_bins2 = np.max(samples_mean2) ## plt.figure(figsize=(14,5)) plt.hist(samples_mean2,n_bins2) plt.xlabel('$n$') plt.ylabel('Frequency of $N = n$') plt.title('Poisson distribution') plt.show()

mean=4 samples_mean4 = np.random.poisson(mean, 10000) n_bins4 = np.max(samples_mean4) ## plt.figure(figsize=(14,5)) plt.hist(samples_mean4,n_bins4,density=True) plt.xlabel('$n$') plt.ylabel('Frequency of $N = n$') plt.title('Poisson distribution') plt.show()

mean=6 samples_mean6 = np.random.poisson(mean, 10000) n_bins6 = np.max(samples_mean6) ## plt.figure(figsize=(14,5)) plt.hist(samples_mean6,n_bins6) plt.xlabel('$n$') plt.ylabel('Frequency of $N = n$') plt.title('Poisson distribution') plt.show()

n_bins = np.max([n_bins2, n_bins4,n_bins6]) plt.figure(figsize=(14,5)) plt.hist([samples_mean2, samples_mean4,samples_mean6],n_bins) plt.xlabel('$n$') plt.ylabel('Frequency of $N = n$') plt.title('Poisson distribution') plt.show()

#time periods size=10000 #Arrival mean=1 arrival = np.random.poisson(mean, size) #Service mean=1.5 service = np.random.poisson(mean, size) # queue = np.zeros(size+1) for i in range(size): if ((arrival[i] - service[i]) + queue[i]) > 0: queue[i+1] = max(0,(arrival[i] - service[i]) + queue[i]) plt.figure(figsize=(14,5)) #plt.plot( arrival, 'o',label='Arrival') #plt.plot( service, 'x',label='Sevice') plt.plot( queue, 'ok') plt.xlabel('$n$') plt.ylabel('Size') plt.title('Queue') #plt.ylim([0, 1]) #plt.xlim([0, 10]) plt.grid(True) #grid plt.show() print('Worst delay:', np.max(queue))

#time periods size=10000 #Arrival mean=1.5 arrival = np.random.poisson(mean, size) #Service mean=1 service = np.random.poisson(mean, size) # queue = np.zeros(size+1) for i in range(size): if ((arrival[i] - service[i]) + queue[i]) > 0: queue[i+1] = max(0,(arrival[i] - service[i]) + queue[i]) plt.figure(figsize=(14,5)) #plt.plot( arrival, '--o',label='Arrival') #plt.plot( service, '-x',label='Sevice') plt.plot( queue, 'ok') plt.xlabel('$n$') plt.ylabel('Size') plt.title('Queue') #plt.ylim([0, 1]) #plt.xlim([0, 10]) plt.grid(True) #grid plt.show() print('Worst delay:', np.max(queue))

#time periods size=50000 #Arrival mean=1 arrival = np.random.poisson(mean, size) #Service mean=1 service = np.random.poisson(mean, size) # queue = np.zeros(size+1) for i in range(size): if ((arrival[i] - service[i]) + queue[i]) > 0: queue[i+1] = max(0,(arrival[i] - service[i]) + queue[i]) plt.figure(figsize=(14,5)) #plt.plot( arrival, '--o',label='Arrival') #plt.plot( service, '-x',label='Sevice') plt.plot( queue, 'ok') plt.xlabel('$n$') plt.ylabel('Size') plt.title('Queue') #plt.ylim([0, 1]) #plt.xlim([0, 10]) plt.grid(True) #grid plt.show() print('Worst delay:', np.max(queue))