Continuous Random Variables

==========

1. Introduction

Call X *continuous* if P[X in A] = integral_A f(x)dx for some function f(x)>0,
called the DENSITY (pdf, for probability density function).  Notice that:
   a) 0 <= f(x) for all x
   b) integral f(x)dx = 1

   *) F(b) = P[X <= b] = integral [f(x): -infinity < x < b] -> f(x) = F'(x)
   *) f(x) CAN be bigger than 1.... can even be infinite!
      Example:  X = U^2 -> F(x) = P[X<x] = P[U<x^.5] = x^.5 -> f(x) = 1/2sqrt(x)

Example:  Lifetime of radio tube:  100/x^2, x>100; 0, x<= 100

PROBLEM:  a)  P[X < 150] = ???                             (1/3)
          b)  P[ 2 of 5 such tubes fail in 150 hrs ] = ??? (5:2)(1/3)^2(2/3)^3=80/243

2. Expectation & Variance of a Continuous Random Variable

   E[g(X)] = sum{ g(x) P[X=x] }    for discrete variables
           = int{ g(x) f(x) dx }   for continuous variables

   In particular, the mean and variance are:

      mu  = E[X] = int{ x f(x) dx } (if it exists!) and
      var = V[X] = int{ (x-mu)^2 f(x) dx } = int{ x^2 f(x) dx } - mu^2


3. The Uniform Random Variable
                                              /   1  \
                                             |  ----- |    if a < x < b
   Uniform on range alpha < X < beta:  f(x) = \  b-a /

                                               (  0  )     if x<a or x>b

   CDF:  F(x) = 0, x<a; (x-a)/(b-a), a<x<b; 1, x>b.

   Mean:  mu = (a+b)/2   Var:  (b-a)^2/12

4. Normal Random Variables
   *) Table, page 214:  F[x], x = 0.00(.01)3.49, P=0.5000...0.9998
   1. The Normal Approximation to the Binomial Distribution
      Cool pix on p.220: exact probs are BINOMIAL, n=10, p=.7:

      0.30 |
      0.25 |                                     .
      0.20 |                                .    :    :
      0.15 |                                :    :    :
      0.10 |                                :    :    :    :
      0.05 |                           :    :    :    :    :    .
      0.00 |__,____,____:____:____:____:____:____:____:____:____:___
              0    1    2    3    4    5    6    7    8    9   10

Normal approx has mean mu = 10*.7 = 7 and sdev sqrt(10*.7*.3) = 1.44914

P[ <= 5 Successes ] = 

     0.0000059049 + 0.0001377810 + 0.0014467005 + 0.0090016920 + 0.0367569090
   + 0.1029193452 = 0.1502683326, or approximately

Phi( (5.5 - 7.0) / 1.44914 ) = Phi ( -1.0351) = 1-Phi(1.0351) =
             = 1 - (.8485+.8508)/2 = 1 - 0.8496 = 0.1504  (not bad!)

   EXAMPLE:  What is the chance of 50 Heads in 100 tosses of fair coin?
   / 100 \        50        50        12611418068195524166851562157
= |       | (0.50)    (0.50)     =   ------------------------------ = 0.0795892...
   \  50 /                           158456325028528675187087900672

  OR, APPROXIMATELY,

= P[ (49.5 - 50)/sqrt(100*.5*.5) < (X-mu)/sqrt(Var) < (50.5 - 50)/sqrt(100*.5*.5)]
= P[ -0.1 < Z < +0.1 ] approx =  2*(0.5398 - 0.5) = 2*(0.0398) = 0.0796

5. Exponential Random Variables  (consider doing hazard FIRST)

        f(x) = lam * e^(-lam * x) ,  x > 0

   P[ T > x ] = exp(-lam * x), x>0

   P[ T < x + eps | T >= x ] = 1 - exp(-lam * (x+eps)) / exp(-lam * x)) / 
                             = 1 - exp(-lam * eps)
                             = lam*eps - (lam*eps)^2/2 + ...
                        approx lam*eps

   ==> lam is *hazard* = death rate; for exponential distribution (ONLY) this
       is constant.

   1. Hazard Rates
                          _
      Survivor function:  F(t) = 1 - F(t) = P[X > t] (optimistic view...)
                                                _
      Hazard:  lambda(t) = f(t)/(1-F(t)) = f(t)/F(t)
      _
      F(x) = exp(-int(0:x) lambda(t) dt)

      Example:  lambda(t) = b*t -->  F(x) = 1 - exp(-b*x^2/2), x>0 ("Rayleigh")
                                     f(x) = b * x * exp(-b*x^2/2),  x>0
                lambda(t) = a   -->  F(x) = 1 - exp(-a*x), x>0  ("Exponential")
                                     f(x) = a * exp(-a*x), x>0

6. Other Continuous Distributions
   1. The Gamma Distribution:   Length of time to catch t fish @ lam/hr avg rate
   2. The Weibull Distribution: Lifetimes: 1-F(t) = exp(-((x-v)/alpha)^beta
   3. The Cauchy Distribution:  Like a normal but much flatter... no mean, var.
   4. The Beta Distribution     Uncertain probability 0<P<1

7. The Distribution of a Function of a Random Variable
      Change-of-variables: Y = g(X) -> f_Y(y) = SUM {f_X(x)/|g'(x)| : g(x)=y}
      Discrete:                        p_Y(y) = SUM {p_X(x)         : g(x)=y}
           Note: something new happened because of Jacobian.

================