Another service from Omega

Joint cdf, pdf, etc..


*****

Definitions

Let X and Y be random variables defined on the same random experiment. We define:

Joint cdf of X and Y




F(s,t) = P([X l.e. s] and [Y l.e. t])

It follows from this definition that,

F(infinity,t) = P(Y l.e. t) = Fy(t)



F(s,infinity) = P(X l.e. s) = Fx(t)


Joint pdf of X and Y




f(s,t) ds dt = P(s<X<s+ds, t<Y<t+dt)

It follows from this definition that,

> f is obtainable from F by taking derivatives

                                        2
                                       d
                          f(s, t) = ----- F(s, t)
                                    dt ds
> F is obtainable from f by integration
                             t           s
                            /           /
                           |           |
              F(s, t) =    |           |     f(x, y) dx dy
                           |           |
                          /           /
                            -infinity   -infinity

Problem

Let (X,Y) be uniformly distributed over the unit circle centered at the origin. Compute the joint pdf, joint cdf, marginal density of X (i.e. fx(s)) and the marginal cdf of Y (i.e. Fy(t)).

Solution:


The joint pdf is constant over the circle and since the total area of the circle is Pi we get that for (x,y) inside the circle,

> f is

                                              1
                              f := (x, y) -> ----
                                              Pi

The joint cdf is computed from f by integration. When (s,t) is a point inside the circle the answer will depend on which cuadrant (s,t) is.

> when (s,t) is in the III or IV cuadrant we have,

                                s    t
                               /    /
                              |    |              1
                   F(s, t) =  |    |             ---- dy dx
                              |    |              Pi
                             /    /
                               -1          2 1/2
                                    -(1 - x )
> integrating over y,
                                    s
                                   /            2 1/2
                                  |   t + (1 - x )
                       F(s, t) =  |   --------------- dx
                                  |         Pi
                                 /
                                   -1

and computing this integral (e.g. with MAPLE)

> F(s,t) = int(int(f(x,y),y=-sqrt(1-x^2)..t),x=-1..s);

                                        2 1/2
                      4 t s + 2 s (1 - s )    + 2 arcsin(s) + 4 t + Pi
        F(s, t) = 1/4 ------------------------------------------------
                                             Pi

The other two simmilar cases: i.e. when (s,t) is in the II or in the I quadrant, are left as exercises. HINT: draw the pictures of the regions.

Now, is this all?
no-no. There is still one more case! To find it try to compute F(0.9,0.8) or F(1/2,1). Actually by thinking about this case you would find the answer to the second question:

The marginal cdf of Y:


This is given by FY(t) where,

> FY(t) = 2*int(int(1/Pi,x=0..sqrt(1-y^2)),y=-1..t);

                                      2 1/2
                            2 t (1 - t )    + 2 arcsin(t) + Pi
                FY(t) = 1/2 ----------------------------------
                                            Pi

where t varies from -1 to 1.

The marginal pdf of X:


By symmetry FY = FX. Thus, the marginal pdf of X is just the derivative w.r.t. t of FY(t) above. This is

> given by

                                           2
                            2 1/2         t              2
                    2 (1 - t )    - 2 ----------- + -----------
                                            2 1/2         2 1/2
                                      (1 - t )      (1 - t )
                1/2 -------------------------------------------
                                        Pi

Done!


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Fri Sep 11 16:20:15 EDT 1998