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

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

     --->  f(x) = (d/dx) F(x)           [ Be careful about RANGES ]

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         <-- IMPORTANT

                         /           0, -oo < x <= a 
                 F(x) = <  (x-a)/(b-a),   a < x <= b
                         \           1,   b < x < oo   

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

                                           -x^2 / 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 unbounded pdf:  X = U^2:
       0 < x < 1 => F(x) = P[ U^2 <= x ]
                         = P[ U   <= sqrt(x) ]
                         = x^{1/2}
                 => f(x) = 1/2sqrt(x) -> oo as x -> 0.
       BUT:  int_{a,b} f(x) dx = sqrt(b)-sqrt(a) < oo.

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 if c < 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

-----------------------------------
Beta Distribution:

  Fix numbers a>0, b>0.  For 0<x<1, set

                      a-1      b-1
      f(x) = const * x    (1-x)

  One can show (but it's not worth our time, imho) that
  the constant must be the fraction

                     Gamma(a+b)
         const = ------------------
                  Gamma(a)*Gamma(b)

  Special case:  a=b=1 -> Uniform
                 b=1   -> F(x) = x^a, 0<x<1
                 a=1   -> F(x) = 1-(1-x)^b, 0<x<1

  These are widely used for modeling uncertain probabilities
  or proportions, and also come up (as we'll see) for order
  statistics.
-----------------------------------

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,...
---------
Relations b/t Dist'ns:

1.  Poisson - Exponential - Gamma:

    Usual assumptions w/ lambda fish per hour;

    Set T_k = time of k'th catch, N_t = # of fish caught in t hrs.

    N_t has Po( mu ) dist'n w/mu = lambda * t; use this to derive
    dist'ns of T_k via e.g.

        { T_1  > t }  =  { N_t = 0 }

        { T_1 <= t }  =  { N_t >= 1 }
        { T_k <= t }  =  { N_t >= k }  w/prob \sum_k^oo P[ N_t=j ]

    Take derivs. to find Gamma, relate to Expo.

----------
2.  Binom - Uniform - Beta - Geometric - Neg Binom

    Let U.1 U.2 ... be iid Un(0,1).

    For fixed n, let X_p be # of U.j's <= p; Bi(n,p) dist'n.

    Of these let V.1 V.2 ... V.n be 1'st, 2'nd, ..., n'th smallest.

    P[ V.k <= p ] = Pr[ X_p >= k ] = sum_k^n P[ X_p=j ]

    ->  pdf for V.k  { Order Statistics -> Beta dist'n }

        F'(p) = sum_k^oo (n:j) (d/dp) p^j (1-p)^(n-j)
              = k(n:k) p^(k-1) (1-p)^(n-k) ~ Be(k,n+1-k)
        E[ V.k ] = k/(n+1)

    Let X_r be # of indices j before r'th U.i <= p; then

       P[ X_r = k ] = P[ r-1 S's and k F's in first k+r-1, then one S ]
                    = (k+r-1:k) p^(r-1) q^k p           k=0,1,2,...
                    = (r*(r+1)*...*(r+k-1)/k!  p^r q^k
                    = Gamma(r+k)/[Gamma(r) k!] p^r q^k