m-procedures for a pair of simple random variables

We examine, first, calculations on a pair of simple random variables X,YX,Y, considered jointly. These are, in effect, two components of a random vector W=(X,Y)W=(X,Y), which maps from the basic space Ω to the plane. The induced distribution is on the (t,u)(t,u)-plane. Values on the horizontal axis (t-axis) correspond to values of the first coordinate random variable X and values on the vertical axis (u-axis) correspond to values of Y. We extend the computational strategy used for a single random variable.

First, let us review the one-variable strategy. In this case, data consist of values t_i and corresponding probabilities P(X=ti)P(X=ti) arranged in matrices

To perform calculations on Z=g(X)Z=g(X), we we use array operations on X to form a matrix

which has g(ti)g(ti) in a position corresponding to P(X=ti)P(X=ti) in matrix PXPX.

Basic problem. Determine P(g(X)∈M)P(g(X)∈M), where M is some prescribed set of values.

We extend these techniques and strategies to a pair of simple random variables, considered jointly.

Joint absolutely continuous random variables

In the single-variable case, the condition that there are no point mass concentrations on the line ensures the existence of a probability density function, useful in probability calculations. A similar situation exists for a joint distribution for two (or more) variables. For any joint mapping to the plane which assigns zero probability to each set with zero area (discrete points, line or curve segments, and countable unions of these) there is a density function.

Definition

If the joint probability distribution for the pair {X,Y}{X,Y} assigns zero probability to every set of points with zero area, then there exists a joint density function fXYfXY with the property

We have three properties analogous to those for the single-variable case:

At every continuity point for fXYfXY, the density is the second partial

Now

A similar expression holds for FY(u)FY(u). Use of the fundamental theorem of calculus to obtain the derivatives gives the result

Marginal densities. Thus, to obtain the marginal density for the first variable, integrate out the second variable in the joint density, and similarly for the marginal for the second variable.

Figure 2: Distribution for Example 4.

Figure 3: Marginal distribution for Example 5.

Discrete approximation in the continuous case

For a pair {X,Y}{X,Y} with joint density fXYfXY, we approximate the distribution in a manner similar to that for a single random variable. We then utilize the techniques developed for a pair of simple random variables. If we have n approximating values t_i for X and m approximating values u_j for Y, we then have n·mn·m pairs (ti,uj)(ti,uj), corresponding to points on the plane. If we subdivide the horizontal axis for values of X, with constant increments dxdx, as in the single-variable case, and the vertical axis for values of Y, with constant increments dydy, we have a grid structure consisting of rectangles of size dx·dydx·dy. We select t_i and u_j at the midpoint of its increment, so that the point (ti,uj)(ti,uj) is at the midpoint of the rectangle. If we let the approximating pair be {X*,Y*}{X*,Y*}, we assign

As in the one-variable case, if the increments are small enough,

The m-procedure tuappr calls for endpoints of intervals which include the ranges of X and Y and for the numbers of subintervals on each. It then prompts for an expression for fXY(t,u)fXY(t,u), from which it determines the joint probability distribution. It calculates the marginal approximate distributions and sets up the calculating matrices t and u as does the m-process jcalc for simple random variables. Calculations are then carried out as for any joint simple pair.

>> tuappr
Enter matrix [a b] of X-range endpoints  [0 1]
Enter matrix [c d] of Y-range endpoints  [0 1]
Enter number of X approximation points  200
Enter number of Y approximation points  200
Enter expression for joint density  3*(u <= t.^2)
Use array operations on X, Y, PX, PY, t, u, and P
>> M = (t <= 0.8)&(u > 0.1);
>> p = total(M.*P)          % Evaluation of the integral with
p =   0.3355                % Maple gives 0.3352455531

The discrete approximation may be used to obtain approximate plots of marginal distribution and density functions.

Figure 4: Marginal density and distribution function for Example 7.

>> tuappr
Enter matrix [a b] of X-range endpoints  [-1 1]
Enter matrix [c d] of Y-range endpoints  [0 1]
Enter number of X approximation points  400
Enter number of Y approximation points  200
Enter expression for joint density  3*u.*(u<=min(1+t,1-t))
Use array operations on X, Y, PX, PY, t, u, and P
>> fx = PX/dx;                % Density for X  (see Figure 4)
                              % Theoretical (3/2)(1 - |t|)^2
>> fy = PY/dy;                % Density for Y
>> FX = cumsum(PX);           % Distribution function for X (Figure 4)
>> FY = cumsum(PY);           % Distribution function for Y
>> plot(X,fx,X,FX)            % Plotting details omitted

These approximation techniques useful in dealing with functions of random variables, expectations, and conditional expectation and regression.