1) The document discusses univariate distribution relationships and provides code examples to generate and plot Bernoulli, binomial, and normal distributions from random variable samples. 2) The code generates random variable samples from Bernoulli, binomial, and normal distributions with varying parameter values and plots the empirical distributions alongside the theoretical distributions. 3) Confidence intervals for the normal distribution are also calculated and printed based on the sample size, probability, and theoretical normal distribution parameters.

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5. Univariate Distribution Relationships
Lawrence M. LEEMIS and Jacquelyn T. MCQUESTON
http://www.math.wm.edu/~leemis/2008amstat.pdf

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9. #
#
p = 0.7
trial_size = 10000
rd.seed(71) #
data = st.bernoulli.rvs(p, size=trial_size)
#
#
p = 0.7
trial_size = 10000
rd.seed(71)
#
data = st.bernoulli.rvs(p, size=trial_size)
h = plt.hist(data, normed=True, color=red, label="rvs", alpha=0.8)
width = np.diff(h[1]).mean()
#
plt.bar([0.1, 0.9], [(1-p)/width, p/width], width=width, color="b", label="Theoretical", alpha=0.8)
plt.legend(loc="best")
plt.show()

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12. #
def gen_bern_var(p, n):
return(np.sum(st.bernoulli.rvs(p, size=n)))
#
p = 0.7
trial_size = 100000
set_size = 30
rd.seed(71)
result = [gen_bern_var(p, set_size) for _
in range(trial_size)]
hist_range = np.arange(np.min(result), np.max(result)+1)
xx = np.arange(0, set_size)
plt.figure(figsize=(10,6))
h = plt.hist(result, normed=True, bins=hist_range, color=red, label="rvs")
plt.bar(xx+0.5, st.binom.pmf(xx, n=set_size, p=p), width=1, alpha=0.5, label="Theoretical")
plt.legend(loc="best") plt.title(f"binomial distribution trial_size={trial_size},
set_size:{set_size}, p:{p}")
plt.show()
n_success: 4655, trial_size:100000
empirical:0.04655, theoretical:0.04644

13. https://www.kaggle.com/kenmatsu4/basket-shot-log-binom
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https://www.kaggle.com/dansbecker/nba-shot-logs

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17. https://www.kaggle.com/kenmatsu4/basket-shot-log-poisson
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18. https://www.kaggle.com/dansbecker/nba-shot-logs

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21. #
set_size = 20000 ; p = 0.1
trial_size = 50000
result = [gen_bern_var(p, set_size)
for _ in range(trial_size)]
xx = np.arange(np.percentile(result, 0.01),
np.percentile(result, 99.99))
# confidence interval
ci = st.norm.interval(0.95, loc=set_size*p,
scale=np.sqrt(set_size*p*(1-p)))
print(f"lower:{ci[0]}, upper:{ci[1]}")
plt.figure(figsize=(10,6))
plt.plot(xx, st.norm.pdf(xx, set_size*p, np.sqrt(set_size*p*(1-p))),
alpha=0.8, label="Theoretical")
plt.hist(result, normed=True, bins=30, color=red, label="rvs", alpha=0.7)
plt.title(f"normal distribution trial_size={trial_size}, set_size:{set_size}, p:{p}")
plt.legend(loc="best")
plt.show()
0.95 confidence interval
lower:1916.8
upper:2083.2

22. https://www.slideshare.net/matsukenbook/rev012
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27. EOF