probability and statistics course project

1234.pdf

16340238_吳德潤.zip

5.png

5.txt

Run the python code below or refer to 5.png for the plots

Observing how these probability distributions evolve as more measurement values are observed, discuss implications of the obtained results:
  The expected value of μ will increase.
  The variance of μ will decrease.

rm -fv 5.png
cat <<EOF | python3
from matplotlib import pyplot
from numpy import linspace
from scipy import stats
from math import sqrt
σ=10
x=[1.74, 3.37, 2.64, 3.86, 3.00, 1.29, 3.65, 3.50, 2.73, 2.88, 3.13, 2.29, 2.66, 0.61, 2.03, 3.62, 4.61, 2.50, 3.98, 3.69]
g5=lambda n:(σ**2/(1+n*σ**2))*sum(x[:n])
β=lambda n:sqrt(σ**2/(1+n*σ**2))
μ=lambda n:linspace(g5(n)-3*β(n), g5(n)+3*β(n), 100)
# https://stackoverflow.com/a/22276109/8243991
f=lambda n,c:pyplot.plot(μ(n), stats.norm.pdf(μ(n),g5(n),β(n)),c)
f(1,'r')
f(3,'g')
f(5,'b')
f(10,'y')
f(20,'c')
# https://matplotlib.org/3.1.1/gallery/text_labels_and_annotations/titles_demo.html
pyplot.title('With the first (red 1, green 3, blue 5, yellow 10, cyan 20) measurement values')
# https://stackoverflow.com/a/12444777/8243991
pyplot.xlabel('μ')
pyplot.ylabel('pmf')
# https://stackoverflow.com/a/9012749/8243991
tmp=pyplot.gcf()
pyplot.show()
pyplot.draw()
tmp.savefig('5.png')
EOF

6.png

6.txt

Run the python code below or refer to 6.png for the plots

Discuss the impacts of σ^2 (when it increases):
  The expected value of μ will increase.
  The variance of μ will increase.

rm -fv 6.png
cat <<EOF | python3
from matplotlib import pyplot
from numpy import linspace
from scipy import stats
from math import sqrt
print()
x=[1.74, 3.37, 2.64, 3.86, 3.00, 1.29, 3.65, 3.50, 2.73, 2.88, 3.13, 2.29, 2.66, 0.61, 2.03, 3.62, 4.61, 2.50, 3.98, 3.69]
print(len(x))
g5=lambda σ:(σ**2/(1+len(x)*σ**2))*sum(x)
β=lambda σ:sqrt(σ**2/(1+len(x)*σ**2))
μ=lambda σ:linspace(g5(σ)-3*β(σ), g5(σ)+3*β(σ), 100)
# https://stackoverflow.com/a/22276109/8243991
f=lambda σ,c:pyplot.plot(μ(σ), stats.norm.pdf(μ(σ),g5(σ),β(σ)),c)
f(0.1,'r')
f(1,'g')
f(10,'b')
# https://matplotlib.org/3.1.1/gallery/text_labels_and_annotations/titles_demo.html
pyplot.title('With σ = (red 0.1, green 1, blue 10)')
# https://stackoverflow.com/a/12444777/8243991
pyplot.xlabel('μ')
pyplot.ylabel('pmf')
# https://stackoverflow.com/a/9012749/8243991
tmp=pyplot.gcf()
pyplot.show()
pyplot.draw()
tmp.savefig('6.png')
EOF

readme.txt

zip 16340238_吳德潤.zip \
  1234.pdf\
  5.png\
  5.txt\
  6.png\
  6.txt\

https://172.18.166.194
students_prob

https://gist.github.com/Un1Gfn/4785d48f56a1eddb5037324d99bd04cb
okul '/home/darren/Telegram Desktop/中期考核题目.pdf'

(1)
https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

(2)
https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Bivariate_case
Known σX σY μX μY
Unknown ρ

(5)

https://glowingpython.blogspot.com/2011/04/how-to-plot-function-using-matplotlib.html
import pylab
import numpy
x = numpy.linspace(-15,15,100) # 100 linearly spaced numbers
y = numpy.sin(x)/x # computing the values of sin(x)/x
# compose plot
pylab.plot(x,y) # sin(x)/x
pylab.plot(x,y,'co') # same function with cyan dots
pylab.plot(x,2*y,x,3*y) # 2*sin(x)/x and 3*sin(x)/x
pylab.show() # show the plot

cat <<EOF | python3
from numpy import sqrt,pi,exp
from numpy import linspace,concatenate
from pylab import plot,show
π=pi
σ=10
# μ=concatenate((linspace(-100,-10,91),linspace(-9,9,10000),linspace(10,100,91)))
μ=linspace(2.8,3.0,100)
# μ=2.9
x=[1.74, 3.37, 2.64, 3.86, 3.00, 1.29, 3.65, 3.50, 2.73, 2.88, 3.13, 2.29, 2.66, 0.61, 2.03, 3.62, 4.61, 2.50, 3.98, 3.69]
A = - sum( ((xi-μ)**2)/2 for xi in x) - (μ**2)/(2*σ**2) + sum( (xi**2)/(2*(σ**2+1)) for xi in x )
n=len(x)
y=1/sqrt(2*π) * ((sqrt(σ**2+1))**n)/σ * exp(A)
print(y)
plot(μ,y)
show()
quit()
EOF

Integrate
cat <<EOF | python3
# https://stackoverflow.com/a/25047602/8243991
# https://glowingpython.blogspot.com/2011/04/how-to-plot-function-using-matplotlib.html
# https://docs.scipy.org/doc/numpy/reference/generated/numpy.linspace.html
from numpy import sqrt,pi,exp
from numpy import linspace,concatenate
from pylab import plot,show
from scipy.integrate import quad
π=pi
σ=10
x=[1.74, 3.37, 2.64, 3.86, 3.00, 1.29, 3.65, 3.50, 2.73, 2.88, 3.13, 2.29, 2.66, 0.61, 2.03, 3.62, 4.61, 2.50, 3.98, 3.69]
n=len(x)
# y=lambda x:x**2
# x=3
# print(y(x))
# x=4
# print(y(x))
A=lambda μ:- sum( ((xi-μ)**2)/2 for xi in x) - (μ**2)/(2*σ**2) + sum( (xi**2)/(2*(σ**2+1)) for xi in x )
y=lambda B:1/sqrt(2*π) * ((sqrt(σ**2+1))**n)/σ * exp(B)
# print(y(A(2.9)))
print(quad(lambda x:y(A(x)),2.8,3.0))
EOF

1/(2*π*σ)*e^(-(x-μ)^2/2-μ^2/(2*σ^2))
integrate -inf ~ +inf x
e^(-mu^2/(2*sigma^2))/(sqrt(2)*sqrt(pi)*sigma)