Pitman MTH 135/ STA 104 Probability Week 8

Change of Variables, Hazard, & Survival


   P[ T <= x + eps | T > x ] = 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
      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.