Question

python coding please

1.2 Sum of the Independent Random Variables Consider a set of 'n random variables XI,Xy . . . Х,, . Let's define the random variable Y as the stinmation of all X, variables: A) For the case m 10 and Xis being independent uniform variables in the interval -0.5,0.5, generate 100,000 samples of Y. Use the discretization technique from the previous section for the [-5,5 interval and plot the pmf of Y B) Now increase m to 100 with Xis being uniform variables in the interval (-0.15,0.15, rpeat the previous part C) Increase m to 1000 with Xis being uniform variables in the nterval [-0.05,0.05), repeat the previous part. What does the distribution of Y look like as you increase m? Compare to the distributions introduced in question D) Repeat the experiments A, B, and C, this time with Xis being independent exponential variables, with rate equal to λ. For m = 10 set λ = 3 for m = 100 set λ = 10, and for m 1000 set λ = 30 Choose the discretization interval properly so almost all of the Y samples are accounted for (It won't be necessarily equal to (-5,5) you saw before). Compare to the distributions from question .1

Answer 1

import numpy as np
import matplotlib.pyplot as plt

def get_Y(m,x_min,x_max):
Y = np.array([]);
for i in range(100000):
X = np.random.uniform(x_min,x_max,m)
Y = np.append(Y,np.sum(X))
return Y

def get_y_exponential(m,lambd):
Y = np.array([]);
for i in range(100000):
X = np.random.exponential(lambd,size=m)
Y = np.append(Y,np.sum(X))
return Y

def get_bins_and_prob(RV,N):
f,b = np.histogram(RV,bins=N) # f=frequency, b=bins
sumf = float(f.sum())
p = f/sumf #probability
return b,p
#print(probs.sum()) # 1.0

def get_pmf(RV,N):
b,probs = get_bins_and_prob(RV,N)
b1 = b[1]
b0 = b[0]
b_1 = b[:-1]
bins = 0.5*(b1-b0) + b_1
plt.figure
plt.bar(bins,probs, 1.0/N)
plt.show()

nbins = 60

# uniform random variable

#A
y = get_Y(10,-0.5,0.5)
get_pmf(y,nbins)

#B
y = get_Y(100,-0.15,0.15)
get_pmf(y,nbins)

#C
y = get_Y(1000,-0.05,0.05)
get_pmf(y,nbins)

#D

#exponentials random variable
y = get_y_exponential(10,3)
get_pmf(y,nbins)

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