def SimulateSample(lam=2, n=10, m=1000):
"""Sampling distribution of L as an estimator of exponential parameter.
lam: parameter of an exponential distribution
n: sample size
m: number of iterations
"""
def VertLine(x, y=1):
thinkplot.Plot([x, x], [0, y], color='0.8', linewidth=3)
estimates = []
for j in range(m):
xs = np.random.exponential(1 / lam, n)
lamhat = 1 / np.mean(xs)
estimates.append(lamhat)
stderr = RMSE(estimates, lam)
print('standard error', stderr)
cdf = thinkstats2.Cdf(estimates)
ci = cdf.Percentile(5), cdf.Percentile(95)
print('confidence interval', ci)
VertLine(ci[0])
VertLine(ci[1])
# plot the CDF
thinkplot.Cdf(cdf)
thinkplot.Save(root='estimation2',
xlabel='estimate',
ylabel='CDF',
title='Sampling distribution')
return stderr
def Estimate2(n=7, m=100000):
"""RMSE for biased and unbiased estimators of population variance.
n: sample size
m: number of iterations
"""
mu = 0
sigma = 1
estimates1 = []
estimates2 = []
for _ in range(m):
xs = [random.gauss(mu, sigma) for i in range(n)]
biased = np.var(xs)
unbiased = np.var(xs, ddof=1)
estimates1.append(biased)
estimates2.append(unbiased)
print('Experiment 2')
print('RMSE biased', RMSE(estimates1, sigma**2))
print('RMSE unbiased', RMSE(estimates2, sigma**2))
def SimulateManyGames(lam, iters=1000000):
lam_est = []
for _ in np.arange(iters):
lam_est.append(SimulateGame(lam))
print('Mean Error =', MeanError(lam_est, lam))
print('RMSE =', RMSE(lam_est, lam))
lam_cdf = thinkstats2.Cdf(lam_est)
ci = lam_cdf.Percentile(5), lam_cdf.Percentile(95)
lam_pmf = thinkstats2.Pmf(lam_est)
thinkplot.Cdf(lam_cdf)
thinkplot.Plot([ci[0], ci[0]], [0, 1], linewidth=2, color='0.8')
thinkplot.Plot([ci[1], ci[1]], [0, 1], linewidth=2, color='0.8')
thinkplot.Config(xlabel='Goals per game', ylabel='CDF', legend=False)
def SampleDistrPLot(estimates, n, lam):
label = 'n=%d' % n
cdf = thinkstats2.Cdf(estimates, label=label)
conf_int = cdf.Percentile(5), cdf.Percentile(95)
stderr = RMSE(estimates, lam)
print('n=', n, 'Std Error=', stderr, 'Conf Int=', conf_int)
thinkplot.Cdf(cdf)
thinkplot.Plot([conf_int[0], conf_int[0]], [0, 1],
color='0.8',
linewidth=2)
thinkplot.Plot([conf_int[1], conf_int[1]], [0, 1],
color='0.8',
linewidth=2)
def Estimate4(lam=2, m=1000000):
estimates = []
for i in range(m):
L = SimulateGame(lam)
estimates.append(L)
print('Experiment 4')
print('rmse L', RMSE(estimates, lam))
print('mean error L', MeanError(estimates, lam))
pmf = thinkstats2.Pmf(estimates)
thinkplot.Hist(pmf)
thinkplot.Show()