Pitman MTH 135/ STA 104 Probability Week 7
Read:  Pitman sections 4.1--4.3

Density Functions, CDF's

* Probability Density Functions

* Exponential and Gamma Distributions

* CDF's
     Continuous Random Variables

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1. Introduction

Recall that the "Cumuilative Distribution Function" or "CDF" is:

                F(x) = P[X <= x]

It's defined for all -oo < x < oo, and satisfies 2 rules:

       a) x < y  =>  F(x) <= F(y)        [ note < and <= signs ]
       b) LIM (F(x): x-> -oo) = 0, LIM (F(x): x-> +oo) = 1
       c) F might not be continuous, but F(x) = LIM {F(x+eps), eps>0}

E.G.:  Binomial, Exponential, Geometric, Normal, Uniform


Call X *continuous* if CDF F(x) = P[X <= x] is a *continuous* function of x

In this course every such F has a derivative f=F' at almost every x and

                F(x) = int{ f(t): -oo < t <= x }

so for any interval A = (a,b] (or any other set A),

              Pr[ X in (a,b] ] = F(b) - F(a)
                               = INT_A { f(x) dx } 

Notice that the "density function" (or "pdf") f satisfies:

   a) 0 <= f(x) for all x
   b) INT { f(x)dx } = 1

   *) F(b) = P[X <= b] = integral [f(x): -oo < x <= b] -> f(x) = F'(x)
   *) f(x) need NOT be continuous;
   *) 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 old-fashioned 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.
   Note: pdf f(x) is indeterminant at x=a, x=b (17? 42?), but
         CDF F(x) is well-defined everywhere ( F(a)=0, F(b)=1 )

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


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 ] = P[ x < T <= x + eps ] / P[ x < T ]
                             = 1 - exp(-lam * (x+eps)) / exp(-lam * x)
                             = 1 - exp(-lam * eps)
                             = lam*eps - (lam*eps)^2/2 + ...
                             = lam*eps + o(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
      Monotone incr:

      F_Y ( g(x) )  =  F_X( x )

      f_Y ( g(x) ) * g'(x) = f_X( x )

      f_Y ( y ) = { f_X( x ) / g'(x) } | g(x) = y
               [ picture illustrating role of g' ]

      Monotone decr;  non-monotone;  

      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: Jacobian only needed for continuous variables [chain rule].

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