Question

python C-E please

C) Generate 1,000, 000 samples from the random variable X of part B. Estimate the empirical mean of X. Plot the pmf of the samples of X. Now suppose you know that you have already played the wheel a few times (say t 3 times), and you have not won yet. Let's define Y:= X-3 for all X> 3. D) Of the samples generated in part C, keep all the samples greater than t 3 and discard the ones that are smaller or equal to t= 3. Then subtract them by 3. These samples now correspond to Y. Estimate the empirical mean of Y. Plot the pmf of the samples of Y. How do they compare to those of X E) Repeat part D for t = 6 and t9. What's your conclusion about the memorylessness of X? As you have seen, no matter how many times you have already played the wheel of fortune, the probability distribution of winning in the future tries remains the same. This random variable X is called Geometric (as you all have got the answer to part B correctly :)). The geometric distribution is the only discrete memoryless distribution. In other words, a discrete random variable is memoryless if and only if it's geometric.

Answer 1

c) The following code gives us the estimate of empirical mean (i.e. the sample mean) and the pmf of the sample:

It gives us the following output:

import numpy as np import matplotlib.pyplot as plt np.random.geometric (p-0.1,size-1000000) X x.mean () mean {mean) print (f'mean = pmf, bins = np.histogram(x,bins=range (1,x.max ()),density=True) plt.plot (bins[:-1],pmf, 'ro') plt.show () -

|mean = 10.017263

0.10 0.08 0.06 0.04 0.02 0.00 40 0 20 60 80 100 120 140

d & e) Now the code below can be used for both part d) and e) to find out the memorylessness from putting the time t=3,6 and 9:

import numpy as np import matplotlib.pyplot as plt x = np.random.geometric(p = 0.1,size 1000000) def f (t) return x[ (x >= t)] - t | y f (3) print (f'mean of Y = pmf, bins plt.plot (bins[:-1],pmf, 'ro') plt.xlabel ('class interval') plt.ylabel('Frequency') plt.show () _ y.mean ()') np.histogram (y,bins range (1,y.max ()),density=True)

it gives us the following output:

mean of Y = 9.00305900281695

0.10 0.08 0.06 0.04 0.02 0.00 0 40 60 80 100 120 Class interval Frequency 20

for part e) also we will get the same pmf from the sample which ensures us about the memoryless property. Their mean also becomes the same as X.