Question

Write a Code in Java or Python, for the following scenario(s):

Consider three six-sided dice, and let random variable Y = the value of the face for each. The probability mass of function of Y is given by the following table:

y	1	2	3	4	5	6	otherwise
P(Y=y)	0.35	0.30	0.25	0.05	0.03	0.02	0

1. Roll the three dice and let random variable X = sum of the three faces. Repeat this experiment 50000 times.

1. Find the simulated probability mass function (pmf) of random variable X.
2. Find the simulated mean of the random variable X.
3. Find the simulated standard deviation of random variable X.
4. Find the simulated P(2 < X < 13).
5. Draw the histogram of the pmf of X.

2. Generate 2000 samples of size 50000 values for the random X from distribution in part1. For each of these 2000 samples calculate the mean of X.

1. Find the simulated probability that the mean is between 6.49 and 6.53 exclusive.
2. Find the simulated mean of the means.
3. Find the simulated standard deviation of the means.
4. Draw the histogram of the means.

Answer 1

A)

import numpy as np
import pylab as plt

N = 5 * 10**4
X = np.random.normal(size=N)

counts, bins = np.histogram(X,bins=50, density=True)
bins = bins[:-1] + (bins[1] - bins[0])/2

print(np.trapz(counts, bins))

B)

a = 1

b = 5 * 10**4

mean = (a+b)/2

print(mean)

C)

a = 1

b = 5 * 10**4

std = (b-a)**2/12

print(std)

D)

a = 2

b = 13

probabilty = 1/(b-a)

print(probability)