P(a < X <= b) =  \int_a^b f(x)dx

F(x) = P[X < x] = \int_{-oo}^x f(t) dt

Expectations:  E[g(X)] = \int g(x) f(x) dx, if \int |g(x)| f(x) dx < oo

        Mean:  E[X] =  \int x f(x) dx
         Var:  V[X] =  \int (x-mu)^2 f(x) dx

Independence:  < wait until we look at joint distributions >

                          const,  a < x <= b
Uniform Dist'n:  f(x) = <                         ( const = 1/(b-a) )
                          0,      other x


        Mean:  E[X] = (a+b)/2
         Var:  V[X] = (b-a)^2/12

                                           -x^2
Std Normal Dist'n:   f(x) = 1/sqrt(2 pi)  e
                                                -(x-mu)^2/2 sig^2
Gen Normal Dist'n:   f(x) = 1/sqrt(2 pi sig^2) e

CLT:                                                    Sn-n mu
      S_n = X1 + X2 + ... + Xn, Var(Xn) < oo ==>  P[ ------------- < z ]
                                                     sqrt(n sig^2)

                                                  -> Phi(z)

---------------------------------

Distn with infinite mean:

       f(x) = 1/x^2, x>1; f(x)=0, x<1

Distn with infinite variance:

       f(x) = 2/x^3, x>1; f(x)=0, x<1


       mu = \int_1^oo x (2/x^3) dx 
          = \int_1^oo 2x^{-2} dx
          = 2

      var = \int_1^oo x^2 (2/x^3) dx  - 4
          = \int_1^oo 1x^{-1} dx - 4
          = oo

----------------------------------

Monte Carlo Integration:

  a)  X_j ~ Unif[a,b]:

      I_n =  ((b-a)/n) \sum  g(X_j)

      E[ I_n ] = (b-a)/n \sum \int_a^b  g(x) f(x) dx
               = (b-a)   \int_a^b g(x) 1/(b-a) dx
               = \int_a^b g(x) dx

      V[ I_n ] = c/n -> 0 as n -> oo


  b)  X_j ~ f(x) :

      set w(x) = [1/f(x)] for every x, and:

      I_n =  (1/n) \sum  g(X_j) w(X_j)

      E[ I_n ] = 1/n \sum \int  g(x) w(x) f(x) dx
               = 1/n \sum \int  g(x) [1/f(x)] f(x) dx
               = \int g(x) dx

      V[ I_n ] = c/n -> 0 as n -> oo

-----------------------------------

Exponential, Gamma Distributions:

  Ex:  f(t) = lam exp( - lam t ),  t > 0


       Pr[ T > s+t | T > t ]  =  Pr[ T > s ]  =  exp(-lam s)


        Radioactive decay?  Poisson Process?  Survival Time?
        Half-life?  Geometric Distribution:

       X = [[ T ]] ->  Pr[ X=x ] = Pr[ x <= T < x+1 ]
                                 = exp(-lam x) - exp(-lam x+1)
                                 = exp(-lam x) [1-exp(-lam)]
                                 = q^x p,  q = exp(-lam), p = 1-q

  Ga:  f(x) = c t^(r-1) exp(-lam t), t > 0  [ c = lam^r/Gamma(r) ]

       where Gamma(r) = \int_0^oo x^{r-1} exp(-x} dx

                      = (r-1)!, when r is an integer 1,2,...