Pitman MTH 135/ STA 104 Probability Week 11

Discrete Conditional Distributions

Let X,Y be min,max of two fair dice; find:

  p(x,y)
  p_2(y)
  p(x|y) = p(x,y) / p_2(y)

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Let X be 1 or 2 with (say) equal probabilities

let Y be Poisson with mean lambda_x (say, x)

Find p(x,y) and p_2(y) and p(x|y).

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Let X,Y ~ f(x,y) = exp(-y), 0<x<y<oo;  find

    p_1(x) =   exp(-x), 0<x<oo

    p_2(y) = y exp(-y), 0<y<oo

    p(x|y) = 1/y,  0<x<y

    p(y|x) = exp(x-y), x<y<oo

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Introduce conditional CDF, conditional expectation, etc.

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4. Conditional Distributions:  Discrete Case

      Simply apply the definition of conditional probability:

                                 P[ X=x && Y=y ]      p(x,y)
      P(x|y) = P[ X=x | Y=y ] = ----------------- = ----------
                                    P[ Y=y ]          p_y(y)

      So, all you need are the joint probability mass function (pmf)
      p(x,y) and the marginal pmf for Y.  Why is it called "Marginal"???
      (draw a table for p(x,y), put row and column sums in the margins)

             p_y(y) = SUM [ p(x,y) : all x ]

      Let N, Q, D be the numbers of nickles, quarters and dollars in a
      pair of coins drawn (without replacement) from a bag containing $2
      of each coin; find the conditional distributions, the conditional
      expectation of Q, given D=1, and the conditional expectation of the
      value (in cents) of the draw, given D=1.  How much more is this than
      the conditional expectation of the value of the draw, given Q=1?  Do
      you expect it to be 75 cents more?  Why?
      (NOTE:  Only need p(q,d) and p_d(d) for d=1)

                  p(q,d)      ????
      p(q | d) = -------- = ----------            (work it out!)
                  p_d(d)      ????

      E[Q | D=1] = SUM  q  p(q | d)      = ???         "  "

5. Conditional Distributions: Continuous Case
                                                  P[X=x,Y=y]        p(x,y)
      Discrete pmf:     p(x|y) = Pr[X=x | Y=y] = -------------- =  --------
                                                    P[Y=y]           p2(y)

                                    f(x,y)
      Continuous pdf:   f(x|y)  =  --------    (think about tiny square)
                                     f2(y)     |
                                               |            _
                        P[X in A | Y in B]:   B            |_|
                                               |
                                         ______|____________ ______________
                                               |            A

      Example:  f(x,y) = (1/y) exp(-y), 0 < x < y < oo  
                       ==> f(y|x) = ??   f(x|y) = ??  (BE CAREFUL about LIMITS)