def binomial_distribution(*args):
try:
n, p = int(args[0]), float(args[1])
except (ValueError, TypeError):
return {'is_valid': False}
if n <= 0 or not 0 < p < 1:
return {'is_valid': False}
t = Symbol('t')
mean, var = binom.stats(n, p)
cf = (1 - p + p * exp(I * t))**n
d_cf = diff(cf, t)
dd_cf = diff(d_cf, t)
mean2 = round(float(lambdify(t, -dd_cf)(0)), 2)
return {
'n': n,
'p': p,
'q': round(1 - p, 2),
'mean': round(float(mean), 2),
'mean2': mean2,
'var': round(float(var), 2),
'g': latex(cf),
'g1': latex(d_cf),
'g2': latex(dd_cf),
'type': type,
'is_valid': True,
}
def testBinom(): # {{{
"""
Binomial Distribution (二项分布 discrete)
二项分布的例子：抛掷10次硬币，恰好两次正面朝上的概率是多少？
事件要么发生, 要么不发生
"""
# 准备数据: 已知 n(伯努利实验次数), p(某件事件发生的概率)
# X轴: n次实验中事件出现k次
# Y轴: 概率
n = 100 # 当n很大(np > 5 && nq > 5) 近似 X ~ N(np, npq)
p = 0.5
xs = np.arange(binom.ppf(0.01, n, p), binom.ppf(0.99, n, p))
# E(X) = np, D(X) = np(1-p)
mean, var, skew, kurt = binom.stats(n, p, loc=0, moments='mvsk')
print("mean: %.2f, var: %.2f, skew: %.2f, kurt: %.2f" %
(mean, var, skew, kurt))
fig, axs = plt.subplots(1, 3)
# 显示pmf
ys = binom.pmf(xs, n, p)
axs[0].plot(xs, ys, 'bo', markersize=5, label='binom pmf')
axs[0].legend()
# 显示cdf
ys = binom.cdf(xs, n, p)
axs[1].plot(xs, ys, 'bo', markersize=5, label='binom cdf')
axs[1].legend()
# 随机变量RVS
data = binom.rvs(n, p, size=1000)
import sys
sys.path.append("../../thinkstats")
import Pmf
pmf = Pmf.MakePmfFromList(data)
xs, ys = pmf.Render()
axs[2].plot(xs, ys, 'bo', markersize=5, label='rvs pmf')
axs[2].legend()
plt.show()
def do_statistics(histogram, latex=False):
data = gen_data(histogram)
emean, evar, eskew, ekurt = binom.stats(TEXT_SIZE * 8, 0.5, moments='mvsk')
estd = binom.std(TEXT_SIZE * 8, 0.5)
mean = s.mean(data)
median = s.median(data)
# We will use sample versions because we don't have all the population
stdev = s.stdev(data)
variance = s.variance(data)
sk = skew(data)
ku = kurtosis(data)
mean_err = abs(emean - mean) / emean
std_err = abs(estd - stdev) / estd
var_err = abs(evar - variance) / evar
kurt_err = abs(ekurt - ku) / ekurt
print("Media:\t\t{}\t\t\t(Err) {}".format(mean, mean_err))
print("Mediana:\t{}".format(median))
print("Desv. Std.:\t{}\t(Err) {}".format(stdev, std_err))
print("Varianza.:\t{}\t(Err) {}".format(variance, var_err))
print("Simetría:\t{}".format(sk))
print("Kurtosis:\t{}\t(Err) {}".format(ku, kurt_err))
if latex:
print("\n\nLaTeX\n")
print(
" &".join("{:>11}".format(x) for x in [
"Muestra", "Media", "Mediana", "Desviación", "Variance",
"Asimetría", "Curtosis"
]), "\\\\ \\hline")
print(
" &".join(
"{:>11.2f}".format(x)
for x in [STUDY_SIZE, mean, median, stdev, variance, sk, ku]),
"\\\\")
print("\n\n")
print(list(enumerate(histogram)))
print("\n\n")
def test_binom(self):
from scipy.stats import binom
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
n, p = 5, 0.4
mean, var, skew, kurt = binom.stats(n, p, moments='mvsk')
x = np.arange(binom.ppf(0.01, n, p), binom.ppf(0.99, n, p))
ax.plot(x, binom.pmf(x, n, p), 'bo', ms=8, label='binom pmf')
ax.vlines(x, 0, binom.pmf(x, n, p), colors='b', lw=5, alpha=0.5)
rv = binom(n, p)
ax.vlines(x,
0,
rv.pmf(x),
colors='k',
linestyles='-',
lw=1,
label='frozen pmf')
ax.legend(loc='best', frameon=False)
self.assertEqual("AxesSubplot(0.125,0.11;0.775x0.77)", str(ax))
def inom():
#二项分布的概率分布图
fig, ax = plt.subplots(1, 1)
n = 100
p = 0.5
#平均值, 方差, 偏度, 峰度
mean, var, skew, kurt = binom.stats(n, p, moments='mvsk')
# print(mean,var,skew,kurt)
#ppf:累积分布函数的反函数。q=0.01时，ppf就是p(X<x)=0.01时的x值。
#这一步是构建累计图的横坐标,当然不这么做
x = np.arange(binom.ppf(0.01, n, p), binom.ppf(0.99, n, p))
# x=np.arange(0,100)
binom.pmf(x, n, p)
# ax.plot(x, binom.pmf(x, n, p),'o')
# plt.show()
#1次试验成功的概率是0.06,没有不合格品概率
print(binom.pmf(0, 5, 0.06))
#5次试验成功的概率是0.06,没一个不合格品概率
print(binom.pmf(1, 5, 0.06))
#有3个及3个一下不合格品
print(binom.cdf(3, 5, 0.06))
from scipy.stats import binom
def ncr(n, r):
f = math.factorial
return f(n) / f(r) / f(n - r)
# setting the values
# of n and p
beta = 10
p = 1.0 / (4.3 * 30 * 24) #-math.exp(-1/(4.3))
print p
# obtaining the mean and variance
mean, var = binom.stats(beta, p)
# list of pmf values
# printing mean and variance
print("mean = " + str(mean))
print("variance = " + str(var))
for beta in [10]:
print "beta={}".format(beta)
for rho in [1, 3, 5, 10]:
availability = 0.0
for k in range(0, beta + 1):
a = 0
if beta - k >= rho:
a = float(ncr(beta - k, rho)) / float(ncr(beta, rho))
failure = binom.pmf(k, beta, p)
availability += a * failure
#程序文件Pex4_3.py
from scipy.stats import binom
n, p=20, 0.8
print("期望和方差分布为：", binom.stats(n,p))
def binomial_distribution(self, DendriteList, ModifyRangeMap):
# setting the values of n and p
# defining the list of k values
n = len(DendriteList)
p = 1 / n
r_values = list(range(n + 1))
# obtaining the mean and variance
mean, var = binom.stats(n, p)
# list of pmf values
# dist = [(binom.pmf(k, n, p)) * n for k in r_values]
dist = []
for k in r_values:
ans = (binom.pmf(k, n, p)) * n
if float(ans) > 0.05:
dist.append(int(math.ceil(ans)))
# printing the table
print(
'\n',
"<--------------- Binomial distribution simulation: --------------->",
'\n')
print(len(dist))
print("k\tp(k)")
# for index_ in r_values:
# print(str(r_values[index_]) + "\t" + "{:.2f}".format(float(dist[index_])))
# # printing mean and variance
# print("mean = " + str(mean))
# print("variance = " + str(var))
parallel_histlist = []
for key, value in ModifyRangeMap.items():
parallel_histlist.append(len(value) + 1)
# print(parallel_histlist)
values_ = histogram(parallel_histlist)
# -------------------fig2 ---------------------
if max(values_) > max(dist[2:len(dist)]):
y_group_range = max(values_)
else:
y_group_range = max(dist[2:len(dist)])
plt.figure(
"Classification of dendritic branch parallel growth vs random simulation"
)
plt.subplot(1, 2, 1)
plt.bar(range(2,
len(values_) + 2),
height=values_,
color="blue",
width=0.35)
# plt.yticks(range(0, max(values_)))
plt.ylim([0, y_group_range])
# plt.yticks(np.arange(0, max(values_),step=2))
plt.xticks(fontsize=20, rotation=45)
plt.yticks(fontsize=20)
plt.ylabel('frequency', fontsize=25, fontweight='bold')
# plt.xlabel('parallel groups', fontsize=10)
plt.xlabel('parallel groups', fontsize=25, fontweight='bold')
plt.title('Measured', fontsize=26)
ax = plt.gca()
ax.xaxis.set_major_locator(MaxNLocator(integer=True))
plt.tight_layout()
plt.subplot(1, 2, 2)
plt.bar(r_values[2:len(dist)],
height=dist[2:len(dist)],
color="blue",
width=0.35)
plt.ylim([0, y_group_range])
# plt.yticks(np.arange(0, max(values_), step=2))
plt.xticks(fontsize=20, rotation=45)
plt.yticks(fontsize=20)
plt.ylabel('frequency', fontsize=25, fontweight='bold')
plt.xlabel('parallel groups', fontsize=25, fontweight='bold')
plt.title('Simulation', fontsize=26)
# plt.yticks(range(0, max(values_)))
ax = plt.gca()
ax.xaxis.set_major_locator(MaxNLocator(integer=True))
plt.tight_layout()
plt.show()
from scipy.stats import binom, expon
import matplotlib.pyplot as plt
import numpy as np
# 离散型的，二项分布
plt.subplot(311)
n = 100
p = 0.5
mean, var, skew, kurt = binom.stats(n, p, moments='mvsk')
# ppf:累积分布函数的反函数。q=0.01时，ppf就是p(X<x)=0.01时的x值
x = np.arange(binom.ppf(0.01, n, p), binom.ppf(0.99, n, p)) # 离散的
# 分布律
plt.plot(x, binom.pmf(x, n, p), 'o')
# 连续型的，指数分布
plt.subplot(312)
plt.title('概率密度')
lambdax = 1 / 5
loc = 0
scale = 1 / lambdax
means, vars, skews, kurts = expon.stats(loc, scale, moments='mvsk')
xx = np.linspace(expon.ppf(0.01, loc, scale), expon.ppf(0.99, loc, scale), 100)
#概率密度
plt.plot(xx, expon.pdf(xx, loc, scale), label='expon')
plt.subplot(313)
plt.title('分布函数')
plt.plot(xx, 1 - np.exp(-lambdax * xx), '*')
plt.show()
print(xx[-1], 1 - np.exp(-lambdax * x[-1]))
from scipy.stats import binom
p = 0.5
k= 100
#'mvsk': 'mean', 'variance'm 'skewness', 'kurtosis'
mean, var, _ , _ = binom.stats(k ,p, moments='mvsk')
print("Mean: %.3f, variance: %.3f" % (mean, var))
## code-runne.run not found issue
from scipy.stats import binom
n, p = 20, 0.8
mean, variance, skewness, kurtosis = binom.stats(n, p, moments='mvsk')
print("所求的数字特征为：", binom.stats(n, p, moments='mvsk'))
# 平均値です。
mu_A = n_A * p_A
mu_B = n_B * p_B
# 標準偏差を計算しましょう。
sigma_A = (n_A * p_A * (1 - p_A))**0.5
sigma_B = (n_B * p_B * (1 - p_B))**0.5
print('プレイヤーAは1試合で、平均{:0.1f}回±{:0.1f}シュートを決めます。'.format(mu_A, sigma_A))
print('\n')
print('プレイヤーBは1試合で、平均{:0.1f}回±{:0.1f}シュートを決めます。'.format(mu_B, sigma_B))
"""
scipyを利用できます
"""
mean, var = binom.stats(n_A, p_A)
print(mean)
print(var**0.5)
"""
確率質量関数も求められます
コインを10回投げて裏と表を出すことを考えてみましょう。
"""
# 10回と、表の確率0.5をセットします。
n = 10
p = 0.5
x = range(n + 1)
# 二項分布の確率質量関数をから、実際の確率を計算できます。
plt.yticks(np.arange(0, 1.1, 0.1))
plt.legend(loc='upper left', shadow=True)
plt.show()
# ### Binomial Distribution
# In[3]:
#Binomial Distribution
from scipy.stats import binom
n, p = 10, 0.4
x = np.arange(binom.ppf(0.01, n, p), binom.ppf(
0.99, n, p)) #Percent Point Function (inverse of cdf — percentiles)
print("Mean : ", binom.stats(n, p, moments='m'))
print("Variance : ", binom.stats(n, p, moments='v'))
print("Prob. Mass Func. : ", binom.pmf(x, n, p))
print("Cum. Density Func.: ", binom.cdf(x, n, p))
CDF = binom.cdf(x, n, p)
fig = plt.figure(figsize=(20, 10))
plt.subplot(221)
plt.plot(x, binom.pmf(x, n, p), 'go', ms=8, label='PMF')
plt.vlines(x, 0, binom.pmf(x, n, p), colors='g', lw=5, alpha=0.5)
plt.xlabel("Sample Space of Binomial Distribution", fontsize=14)
plt.ylabel("PMF", fontsize=14)
plt.title("Probability Distribution of Binomial(n=10,p=0.4) Distribution",
fontsize=16)
plt.xticks(np.arange(0, 8, 1))
plt.show()
#Binomial distribution
#Example1
#https://data-flair.training/blogs/python-probability-distributions/
#Describes variation in the number of one outcome from replicate to replicate
import seaborn as sb
import matplotlib.pyplot as plt
from scipy.stats import binom
n=20
p=0.8
binom.rvs(size=10,n=n,p=p)
moments = binom.stats(n, p, moments='mvsk')
data_binom = binom.rvs(n=20,p=0.8,loc=0,size=1000)
ax = sb.distplot(data_binom,kde=True,color='blue',hist_kws={"linewidth": 25,'alpha':1})
ax.set(xlabel='Binomial', ylabel='Frequency')
plt.show()
print('mean,var,skew,kurt =', moments)
#Example2
#Describes variation in the number of one outcome from replicate to replicate
from scipy.stats import binom
import matplotlib.pyplot as plt
import seaborn as sb
n = 484382 #number of of particular set of trials (e.g. male births)
N = 938223 #total number
ep = (n-1)/(N+2) #estimated probability of occurence of each trial
size = shape of returned array
# Distribucion Binomial usando scipy.stats
from scipy.stats import binom
import numpy as np
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
# Calculamos los primeros momentos:
n, p = 5, 0.4
mean, var, skew, kurt = binom.stats(n, p, moments='mvsk')
# Mostramos el pmf de la variable aleatoria (``pmf``):
x = np.arange(binom.ppf(0.01, n, p),
binom.ppf(0.99, n, p))
ax.plot(x, binom.pmf(x, n, p), 'bo', ms=8, label='pmf binomial')
ax.vlines(x, 0, binom.pmf(x, n, p), colors='b', lw=5, alpha=0.5)
ax.legend(loc='best', frameon=False)
# Comprobar la exactitud del ``cdf`` y ``ppf``:
prob = binom.cdf(x, n, p)
np.allclose(x, binom.ppf(prob, n, p))
# Generamos numeros aleatorios
r = binom.rvs(n, p, size=1000)
plt.show()
import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import binom, skew
np.random.seed(5)
# Binomial Distribution Parameters
n = 10
p = 0.2
# Descriptive Stats for Original Probability Distribution
pop_mean, pop_var = binom.stats(n, p, loc=0, moments='mv')
pop_std = binom.std(n, p)
print('Population Mean: ', pop_mean)
print('Population Variance: ', pop_var)
# Random samples
x = np.arange(0, n + 1)
pmf = binom(n, p).pmf(x)
# Sample statistics
mean_of_means = []
variances = []
skews = []
samp_sizes = [5, 10, 50, 100]
for i, samp_size in enumerate(samp_sizes, start=1):
means = []
for sample in range(10000):
pts = binom.rvs(n, p, size=samp_size)
import matplotlib.pyplot as plt
from scipy.stats import binom
# n is the number of trials
n = int(input("Input the number of trials: "))
# p is the probability of success
p = float(input("Input the success probability: "))
# While probabilities should be <= to 1
while p > 1:
print("p value should be <= 1")
p = float(input("Input the success probability: "))
# defining the list of r values
r_values = list(range(n + 1))
# mean and variance
mean, var = binom.stats(n, p)
# list of pmf values
dist = [binom.pmf(r, n, p) for r in r_values]
# printing the table
print("r\tp(r)")
for i in range(n + 1):
print(str(r_values[i]) + "\t" + str(dist[i]))
# Plot values
plt.bar(r_values, dist)
plt.show()
# Display MEAN, VARIANCE
print("mean = " + str(mean))
print("variance = " + str(var))
