1. Implement a program to compute π value using Monte Carlo simulation method. Use Python 3 and the name pi.py
(a) The program should generate n points to compute π for n ∈ {103,104,105,106}.
(b) You can use math.pi to compute error rates.
(c) Please follow the output format. (Fix precisions using "0:.nf".format) n = 10 ^ 3 pi = 3.096000 error = 1.4513 % n = 10 ^ 4 pi = 3.136800 error = 0.1526 % n = 10 ^ 5 pi = 3.145280 error = 0.1174 % n = 10 ^ 6 pi = 3.140568 error = 0.0326 %
2. Consider Figure 1, and implement a program to answer the query ) by using Gibbs (MCMC) sampling. The program should generate 1,000,000 samples to estimate the probability. Use Python 3 and the name gibbs.py
(a) Show
(b) Show the transition probability matrix Q ∈ R4×4 where qij = transition probability from Si to Sj in Figure 2.
(c) Show the probability of the query
(d) Please follow the output format. (Fix precisions using "0:.nf".format)
Part A. The sampling probabilities
P(C|-s,r) = <..., ...
P(C|-s,-r) = <..., ...
P(R|c,-s,w) = <..., ...
P(R|-c,-s,w) = <..., ...
Part B. The transition probability matrix
S1
S2
S3
S4
S1
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S2
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S3
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S4
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Part C. The probability for the query
P(C|-s,w) = <..., ...
Figure 1: A multiply connected network with conditional probability tables
Figure 2: Possible states diagram
