.. probability lecture

Helpful math
=============

Properties of exponentials
------------------------------
   
   +--------------------------------------+
   | :math:`a^{-n} = \frac{1}{a^{n}}`     |
   +--------------------------------------+

>>> a,n = 2,3
>>> print(a**(-n),1.0/a**n)
0.125 0.125

   +-------------------------------------+
   | :math:`a^{1/n} = \sqrt[n]{a}`       |
   +-------------------------------------+

>>> a,n = 2,3 
>>> print(a**(1/n), np.cbrt(a))
1.2599210498948732 1.25992104989

   +-------------------------------------+
   | :math:`a^{m/n} = \sqrt[n]{a^{m}}`   |
   +-------------------------------------+

>>> a,m,n = 2,4,3
>>> print(a**(m/n), np.cbrt(a**m))
2.5198420997897464 2.51984209979
   
   +-------------------------------------+
   | :math:`a^{mn} = (a^{m})^{n}`        |
   +-------------------------------------+

>>> a,m,n = 2,4,3   
>>>  print(a**(m*n), (a**m)**n)
4096 4096

   +-------------------------------------+
   | :math:`\log(x^{y}) = y \log(x)`     |
   +-------------------------------------+

>>> x,y = 4,2
>>> print(np.log(x**y),y * np.log(x))
2.77258872224 2.77258872224

   +-------------------------------------+
   |  :math:`\log_{a} a^{n} = n`         |
   +-------------------------------------+

>>> x
>>> x
>>> x

   +--------------------------------------+
   | :math:`\log(bc) = \log(b) + \log(c)` |
   +--------------------------------------+

>>> x
>>> x
>>> x

   +--------------------------------------+
   | :math:`\log(\frac{1}{b}) = -log(b)`  |
   +--------------------------------------+

>>> x
>>> x
>>> x

   
Also recall that  :math:`n^{0} = 1, \ 1! = 1, \ 0! = 1`