# integral (expression, variable, range)
integral(exp(-1/2*x^2),x,-oo,oo)
|
|
def f(x,a,b,c,d):
return a*x^3 + b*x^2 + c*x + d
#define vector
var("x a b c d")
#solve equation w/ each constraints
solve([f(0,a,b,c,d)==1/2,
f(1,a,b,c,d)==1/3,
f(2,a,b,c,d)==1/12,
f(3,a,b,c,d)==1/12],
a,b,c,d)
|
|
#plot example
plot (x^3, (x,-1,1), color='red' )
|
|
f(x) = exp(-1*x^2);f
|
|
plot(f,(x,-4,4),color="black")
|
|
# sqrt(pi)로 나누어주어 전체 적분값이 1이 되도록 맞춘다
integral(exp(-1*x^2),x,-oo,oo)
integral(f,x,-oo,oo)
|
|
# 평균=0이므로 분산은
integral(1/sqrt(pi)*f*(x-0)^2, x,-oo,oo)
|
|
# mean=0, var=1 이면서 전체적분값이 1인 pdf로 만들어주어야함
# integral f dx = 1 / integral f*x dx = 0 / integral f*(x-mean)^2 dx = 1
|
|
#mixed normal distribution
def fNormal(x, mu, sd1):
return 1/sqrt(2*pi*sd1^2)*exp(-1/2*((x-mu)/sd1)^2)
def fN_summed(x, mu1, mu2, sd1, sd2):
return fNormal(x, mu1+mu2, sqrt(sd1^2 + sd2^2))
def fN_mixed(x, mu1, mu2, sd1, sd2, p1):
return p1*fNormal(x, mu1, sd1) + (1-p1)*fNormal(x, mu2, sd2)
a = plot(fNormal(x,1,1), (x,-3,3), linestyle=":", color="blue") #dashline
b = plot(fNormal(x,-1,1), (x,-3,3), linestyle="-.", color="red")
c = plot(fN_summed(x, -1,1,1,1), (x,-3,3), linestyle="-", color="black")
show(a+b+c)
|
|
a = plot(fNormal(x,1,1), (x,-3,3), linestyle=":", color="blue") #dashline
b = plot(fNormal(x,-1,1), (x,-3,3), linestyle="--", color="red")
c = plot(fN_mixed(x, -1,1,1,1,1/2), (x,-3,3), linestyle="-", color="black")
show(a+b+c)
|
|
# normal distribution
def f1(x,mu1,sigma1):
return 1/sqrt(2*pi*sigma1^2)*exp(-1/2*((x-mu1)/sigma1)^2)
plot(f1(x,0,1),x,-4,4)
|
|
# Laplace distribution
def f2(x,mu2,sigma2):
return 1/sqrt(2*pi*sigma2^2)*exp(-sqrt(2)*abs((x-mu2)/sigma2))
plot(f2(x,0,1),x,-4,4)
|
|
var("mu sigma")
assume(sigma>0)
integral(x*f1(x,mu,sigma),x,-oo,oo)
|
|
factor(sqrt(pi)*sqrt(1/2)*sqrt(2)*mu/(sqrt(pi*sigma^2)*sqrt(sigma^(-2))))
|
|
|
|
|
|
|
|