3306.sagews

C(n, k) = factorial(n) / (factorial(k) * factorial(n - k))

#задача1

#a)
n = 4; m = 13; m1 = 5; m2 = 8
k = 3
float(C(n,k)*(m1/m)^k*(m2/m)^(n - k))*

\newcommand{\Bold}[1]{\mathbf{#1}}0.140051118658

#b)
k = 2
float(1 - (C(n,k - 2)*(m2/m)^n + C(n,k - 1)*(m1/m)*(m2/m)^(n - k + 1)))

\newcommand{\Bold}[1]{\mathbf{#1}}0.498056790729

#задача2

#a)
float(C(5,2)*0.51^2*0.49^3)

\newcommand{\Bold}[1]{\mathbf{#1}}0.306005049

#b)
float(C(5,0)*0.49^5 + C(5,1)*0.51*0.49^4 + C(5,2)*0.51^2*0.49^3)

\newcommand{\Bold}[1]{\mathbf{#1}}0.4812549994

#c)
float(1 - (C(5,0)*0.49^5 + C(5,1)*0.51*0.49^4 + C(5,2)*0.51^2*0.49^3))

\newcommand{\Bold}[1]{\mathbf{#1}}0.5187450006

#d)
float(C(5,2)*0.51^2*0.49^3 + C(5,3)*0.51^3*0.49^2)

\newcommand{\Bold}[1]{\mathbf{#1}}0.6245001

#задача3

#a)
l=2
t=4
m=3
float((((l*t)^m)/factorial(m)) * e^(-l*t))

\newcommand{\Bold}[1]{\mathbf{#1}}0.0286261442477

#b)
P = 0
for m in range(3):
    P += (((l*t)^m)/factorial(m)) * e^(-l*t)
    print float((((l*t)^m)/factorial(m)) * e^(-l*t)),
print "\n", float(P)

0.000335462627903 0.00268370102322 0.0107348040929 0.013753967744

#c)
float(1 - P)

\newcommand{\Bold}[1]{\mathbf{#1}}0.986246032256

#задача4

#a)
n = 100
k = 50
p = q = 1/2
x =(k - n*p) / sqrt(n*p*q)
f = float((1 / sqrt(2*pi)) * e^(-(x^2)/2))
x, f, float(f / sqrt(n*p*q))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(0, 0.398942280401, 0.0797884560803\right)

#b)
k = 60
x = (k - n*p) / sqrt(n*p*q)
f = float((1 / sqrt(2*pi)) * e^(-(x^2)/2))
x, f, float(f / sqrt(n*p*q))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(2, 0.0539909665132, 0.0107981933026\right)

L = [[float((k - n*p) / sqrt(n*p*q)), float(((1 / sqrt(2*pi)) * e^(-(float((k - n*p) / sqrt(n*p*q))^2)/2)) / sqrt(n*p*q))] for k in range(101)]
line(L)

#задача5

n = 100
k = 50
p = 0.51
q = 0.49
x =(k - n*p) / sqrt(n*p*q)
f = float((1 / sqrt(2*pi)) * e^(-(x^2)/2))
x, f, float(f / sqrt(n*p*q))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(-0.200040012004001, 0.391039564395, 0.0782235591555\right)

#задача6

n = 2100
p = 0.7
q = 0.3

#a)
m1 = 1470; m2 = 1500
x1 = (m1 - n*p) / sqrt(n*p*q)
x2 = (m2 - n*p) / sqrt(n*p*q)
x1, x2, float((1 / sqrt(2*pi)) * integrate(e^(-(x^2)/2), x, x1, x2))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(0.000000000000000, 1.42857142857143, 0.42343627449\right)

#c)
m1 = 0; m2 = 1469
x1 = (m1 - n*p) / sqrt(n*p*q)
x2 = (m2 - n*p) / sqrt(n*p*q)
x1, x2, float((1 / sqrt(2*pi)) * integrate(e^(-(x^2)/2), x, x1, x2))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(-70.0000000000000, -0.0476190476190476, 0.481009925722\right)

#b)
m1 = 0; m2 = 1469
x1 = (m1 - n*p) / sqrt(n*p*q)
x2 = (m2 - n*p) / sqrt(n*p*q)
x1, x2, 1 - _[2]

\newcommand{\Bold}[1]{\mathbf{#1}}\left(-70.0000000000000, -0.0476190476190476, 0.518990074278\right)

#задача7

n = 900
p = 0.9
q = 0.1
m1 = 790; m2 = 830
x1 = (m1 - n*p) / sqrt(n*p*q)
x2 = (m2 - n*p) / sqrt(n*p*q)
x1, x2, float((1 / sqrt(2*pi)) * integrate(e^(-(x^2)/2), x, x1, x2))

\newcommand{\Bold}[1]{\mathbf{#1}}\left(-2.22222222222222, 2.22222222222222, 0.973731708618\right)