Statistical parameters using probability density function

Given probability density function, $p(x)$ ¶

$p = 2x/b^2$ , $0 < x < b$

$\overline{x} = \int\limits_0^b x\, p(x)\, dx = \int\limits_0^b 2x^2/b^2 = \left. 2x^3/3b^2\right|_0^b =2b^3/3b^2 = 2b/3$

median: $\int\limits_0^m p(x)\,dx = 1/2 = \int\limits_0^m 2x/b^2\,dx = \left. x^2/b^2 \right|_0^m = m^2/b^2 = 1/2$ , $m = b/\sqrt(2)$

second moment: $x^{(2)} = \int\limits_0^b x^2\, p(x)\, dx = \int\limits_0^b 2x^3/b^2 = \left. x^4/2b^2\right|_0^b =b^4/2b^2 = b^2/2$

$var(x) = x^{(2)} - \overline{x}^2 = b^2/2 - 4b^2/9 = b^2/18$

from pylab import *
import numpy as np

def p(x,b):
    return 2*x/(b**2)

b = 2
x = linspace(0,b,200)
y = p(x,b)

plot(x,y)
xlabel('$x$')
ylabel('$p(x)$')

# approximate using the numerical integration
print(trapz(y*x,x))
print(2.*b/3)

1.3333501679250523
1.3333333333333333

print(trapz(y*x**2,x))
print(b**2/18)

2.000050503775157
0.2222222222222222

import sympy

sympy.var('x,b,p,m')
p = 2*x/b**2
print (p)

2*x/b**2

sympy.integrate(p*x,(x,0,b))

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sympy.integrate(p*x**2,(x,0,b))

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sympy.integrate(p,(x,0,m))

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sympy.solve(m**2/b**2 - 0.5,m)

[-0.707106781186548*b, 0.707106781186548*b]